Optimality Conditions for Multiobjective Mathematical Programming Problems with Equilibrium Constraints on Hadamard Manifolds

Abstract

In this paper, we consider a class of multiobjective mathematical programming problems with equilibrium constraints on Hadamard manifolds (in short, (MMPEC)). We introduce the generalized Guignard constraint qualification for (MMPEC) and employ it to derive Karush–Kuhn–Tucker (KKT)-type necessary optimality criteria. Further, we derive sufficient optimality criteria for (MMPEC) using geodesic convexity assumptions. The significance of the results deduced in the paper has been demonstrated by suitable non-trivial examples. The results deduced in this article generalize several well-known results in the literature to a more general space, that is, Hadamard manifolds, and extend them to a more general class of optimization problems. To the best of our knowledge, this is the first time that generalized Guignard constraint qualification and optimality conditions have been studied for (MMPEC) in manifold settings.

1. Introduction

It has recently been observed that many practical problems have emerged in several branches of science and engineering, which can be modelled more efficiently on a manifold space than an Euclidean space; see [1,2]. Moreover, there are various advantages to extending and generalizing the optimization methods from Euclidean spaces to manifold spaces. For instance, a constrained optimization problem can be transformed into a much simpler unconstrained problem through the suitable use of the Riemannian geometry. Further, it is possible to get rid of non-convexity in many optimization problems through the introduction of appropriate Riemannian metrics (see, for instance, [3,4]). In manifold settings, the concepts of convex sets and convex functions are generalized to the concepts of geodesic convex sets and geodesic convex functions (see, for instance, [5,6]). The concepts of pseudoconvexity and quasiconvexity in a geodesic sense were introduced by Udrişte [6] in the Riemannian manifolds setting. Recently, many authors have generalized various other notions and concepts of optimization, from Euclidean spaces to Riemannian manifolds; see, for instance, [7,8,9,10,11,12,13,14,15,16,17,18] and the references cited therein.

A mathematical programming problem with equilibrium constraints (MPEC) is an optimization problem with certain complementarity constraints or variational inequality constraints. The initial conceptualization and formulation of (MPEC) is attributed to Harker and Pang [19]. (MPEC) has various real-life applications in several branches of engineering and science, for instance, hydro-economic river basin model problems [20], telecommunication [21], chemical engineering [22], etc. In recent years, several authors have extensively studied such problems; see, for instance, [23,24,25,26,27,28,29] and the references cited therein.

A comprehensive study of (MPEC) was presented by Luo et al. [30]. Feasibility conditions for (MPEC) were discussed by Fukushima and Pang [31]. A necessary optimality condition for a class of MPEC was given by Outrata [32]. Several stationary point concepts were introduced and discussed by Scheel and Scholtes [33]. Chen and Florian [34] derived regularity and optimality conditions for (MPEC) and deduced new constraint qualifications for such problems. Necessary and sufficient optimality conditions for (MPEC) were studied by Ye [35]. Abadie-type constraint qualification and Slater-type constraint qualifications for (MPEC) were provided by Flegel and Kanzow [36]. Further, Flegel and Kanzow [23] established that first-order optimality conditions for (MPEC) can be deduced under the assumption of a Guignard constraint qualification. KKT-type optimality conditions and duality results for (MPEC) were established by Singh and Mishra [28]. Sadeghi-Mobarakeh and Mohsenian-Rad [37] modelled an optimal bidding problem in performance-based regulation markets for a large price-maker regulation resource in terms of MPEC. Wogrin et al. [38] proposed a bi-level model in the form of an (MPEC), with the aim of providing assistance to a generation company making long-term generation capacity investment decisions, considering uncertainty regarding the investments of the other generation companies. The (MPEC) model was employed to study the strategic behaviors of a profit-driven energy hub in the electricity and heating markets against the background of energy system integration by Li et al. [39].

Motivated by the works of [23,28,35,40] and the references cited therein, in this paper, we study a class of multiobjective programming problems with equilibrium constraints on Hadamard manifolds (MMPEC). We introduce the generalized Guignard constraint qualification for MMPEC and employ it to derive the KKT-type necessary optimality criteria for (MMPEC). Further, we derive sufficient optimality criteria for (MMPEC) using geodesic convexity assumptions. We also provide a useful application of such problems in the field of process engineering, and discuss the advantages of modelling such problems on manifolds. Several non-trivial examples are given, which justify and validate the results derived in the paper.

The novelty and contributions of our work are twofold. In the first fold, we extend the corresponding results deduced by [35,40] to a more general space, namely, Hadamard manifolds, and generalize them to a more general class of optimization problems, namely, multiobjective mathematical programming problems with equilibrium constraints. In the second fold, we extend the corresponding optimality conditions given by Singh and Mishra [28] to a wider space, that is, Hadamard manifolds. Moreover, the results of this paper generalize some other well-known corresponding results in Euclidean spaces; see, for instance, [19,23,41]. To the best of our knowledge, generalized Guignard constraint qualification and optimality conditions of (MMPEC) on Hadamard manifolds were considered for the first time in this article.

This paper is organized as follows. Some basic mathematical preliminaries and concepts used in the sequel are discussed in Section 2. In Section 3, we introduce the generalized Guignard constraint qualification on Hadamard manifolds for (MMPEC) and employ it to derive the KKT-type necessary optimality conditions for our considered problem. In Section 4, sufficient optimality conditions for (MMPEC) were derived under the assumptions of geodesic convexity. An interesting application of mathematical programming problems with equilibrium constraints in the field of process engineering is described, and the advantages of modelling such problems on manifolds is discussed in Section 5. Conclusions are drawn in Section 6, together with some directions for future research.

2. Notation and Mathematical Preliminaries

Throughout the article, the standard notation ${\mathbb{R}}^n$ is used to denote the n-dimensional Euclidean space. The set of all natural numbers is denoted by the standard notation $\mathbb{N}$. The symbol ${\mathbb{R}}_{+}^n$ denotes the non-negative orthant of ${\mathbb{R}}^n$, that is,

$${\mathbb{R}}_{+}^n:=\left\{(x_1,x_2,\dots ,x_n):x_k\ge 0,\forall k=1,2,\dots ,n\right\}.$$

The Euclidean inner product on ${\mathbb{R}}^n$ is denoted by the symbol $\langle ·,·\rangle$. For every $u,v\in {\mathbb{R}}^n$, we employ the following notation:

$$\begin{array}{cc}\hfill u& \prec v⟺u_j<v_j,\forall j=1,2,\dots ,n.\hfill \\ \hfill u& ⪯v⟺\left\{\begin{array}{c}{u}_j\le v_j,\mathrm{for}\mathrm{all}j=1,2,\dots ,n;j\ne k;\hfill \\ u_k<v_k,\mathrm{for}\mathrm{at}\mathrm{least}\mathrm{one}k\in \{1,2,\dots ,n\}.\hfill \end{array}\right.\hfill \end{array}$$

The notation $u\nprec v$ (respectively, $u⋠v$) indicates the negation of $u\prec v$ (respectively, $u⪯v$).

Now, we recall some fundamental concepts and definitions of Riemannian and Hadamard manifolds (see, for instance, [42,43,44,45]).

Any topological space $\mathcal{H}$ is defined as a topological manifold of dimension n (in other words, topological n-manifold) if $\mathcal{H}$ is Hausdorff, second-countable, and every element of $\mathcal{H}$ is always contained in a neighborhood that is homeomorphic to an open subset of ${\mathbb{R}}^n$. Any pair $(V,\psi )$, where V is some open set in $\mathcal{H}$, and $\psi$ is a homeomorphism from V to some open set in ${\mathbb{R}}^n$, is referred to as a chart (in other words, a coordinate chart) on the manifold $\mathcal{H}$. For every pair of coordinate charts $(V_{\alpha },{\psi }_{\alpha })$ and $(V_{\beta },{\psi }_{\beta })$, for which $V\alpha \cap V_{\beta }$ is non-empty, the composite map ${\psi }_{\alpha }\circ {\psi }_{\beta }^{-1}:{\psi }_{\beta }(V_{\alpha }\cap V_{\beta })\to {\psi }_{\alpha }(V_{\alpha }\cap V_{\beta })$ is referred to as the transition map (see Figure 1). Any pair of charts $(V_{\alpha },{\psi }_{\alpha })$ and $(V_{\beta },{\psi }_{\beta })$ are referred to as smoothly compatible if either $V_{\alpha }\cap V_{\beta }$ is empty or the transition map ${\psi }_{\alpha }\circ {\psi }_{\beta }^{-1}$ is infinitely continuously differentiable.

Any collection of charts such that the corresponding open sets cover the manifold $\mathcal{H}$ is referred to as an atlas. An atlas $\mathcal{A}$ for $\mathcal{H}$ is said to be smooth if every pair of coordinate charts in $\mathcal{A}$ are smoothly compatible. A smooth atlas $\mathcal{A}$ on $\mathcal{H}$ is referred to be maximal if it is not properly contained in any larger smooth atlas. A maximal smooth atlas on the manifold $\mathcal{H}$ is called a smooth structure. A smooth manifold is a pair ($\mathcal{H},\mathcal{A})$, where $\mathcal{H}$ is some topological manifold and $\mathcal{A}$ is a smooth structure on $\mathcal{H}$.

For an element $p\in \mathcal{H}$, a curve $\zeta :(-ϵ,ϵ)\to \mathcal{H}$ is said to be of class $C^1$ around the element p if $\zeta \left(0\right)=p$, and $\varphi \circ \zeta$ is continuously differentiable for every chart ($U,\varphi$) around the element p. Let ${\zeta }_1,{\zeta }_2:(-ϵ,ϵ)\to \mathcal{H}$ be any two $C^1$ curves about p. Then, ${\zeta }_1$ and ${\zeta }_2$ are said to be equivalent if, and only if, there is some chart ($U,\varphi$) about the point p, such that

$${(\varphi \circ {\zeta }_1)}^{\prime }\left(0\right)={(\varphi \circ {\zeta }_2)}^{\prime }\left(0\right).$$

Any equivalence class of $C^1$ curves through a point $p\in \mathcal{H}$ on $\mathcal{H}$ modulo the equivalence relation defined above is referred to as a tangent vector to $\mathcal{H}$ at the point p.

The set containing all tangent vectors at the point p in $\mathcal{H}$ is termed the tangent space to $\mathcal{H}$ at p and is denoted by the symbol $T_p\mathcal{H}$.

A symmetric and positive definite 2-tensor field $\mathcal{G}$ defined on a smooth manifold $\mathcal{H}$ is called a Riemannian metric. Every Riemannian metric induces an inner product on every tangent space $T_p\mathcal{H}$, and this is denoted by $\mathcal{G}(u,v)={\langle u,v\rangle }_p$ for all $u,v\in T_p\mathcal{H}$. Any smooth manifold with a given Riemannian metric is termed a Riemannian manifold. The exponential map ${exp}_p:T_p\mathcal{H}\to \mathcal{H}$ is defined by ${exp}_p\left(v\right)={\zeta }_{p,v}\left(1\right)$ for any $v\in T_p\mathcal{H}$, where ${\zeta }_{p,v}$ is the geodesic, starting at p with a velocity of v.

A Riemannian manifold $\mathcal{H}$ is defined as a geodesic complete if, for every $x\in \mathcal{H}$, ${exp}_x\left(v\right)$ is defined for element tangent vector $v\in T_x\mathcal{H}$. Any Riemannian manifold that is complete, connected, and has nonpositive sectional curvature, is referred to as a Hadamard manifold.

The following theorem, popularly known as the Hadamard–Cartan theorem, derives from Sakai ([46] Theorem 4.1, p. 221).

Theorem 1.

Let $\mathcal{H}$ be any Hadamard manifold. Then, for every element $p\in \mathcal{H}$, the exponential map ${exp}_p:T_p\mathcal{H}\to \mathcal{H}$ is a diffeomorphism, such that the inverse exponential map ${exp}_p^{-1}:$$\mathcal{H}\to T_p\mathcal{H}$ satisfies ${exp}_p^{-1}\left(p\right)=0_p$. Furthermore, for every element $x\in \mathcal{H}$, there is always some unique minimal geodesic ${\zeta }_{p,x}:[0,1]\to \mathcal{H}$ satisfying ${\zeta }_{p,x}\left(t\right)={exp}_p\left(t{exp}_p^{-1}\left(x\right)\right)$.

The following definition of geodesic convex sets derives from Udrişte ([6], p. 57).

Definition 1.

Let $\mathcal{H}$ be a Riemannian manifold and $S\subseteq \mathcal{H}$. Then, S is referred to as a geodesic convex set in $\mathcal{H}$ if, for every pair of elements, $p,q\in S$, $p\ne q$, and for every geodesic ${\zeta }_{p,q}:$$[0,1]\to \mathcal{H}$ that joins p to q, we have

$${\zeta }_{p,q}\left(t\right)\in S,\forall t\in [0,1].$$

The following definition of geodesic convex functions derives from Rapcsák ([5], Definition 6.1.2, p. 64).

Definition 2.

Let S be a geodesic convex subset of a Riemannian manifold $\mathcal{H}$, and let $g:S\to \mathbb{R}$ be a function on S. Then, the function g is referred to as a geodesic convex function at $y\in S$ if, for every element $z\in S$ and for every geodesic ${\zeta }_{y,z}:[0,1]\to \mathcal{H}$ joining y to z, the following inequality holds:

$$g\left({\zeta }_{y,z}\left(\lambda \right)\right)\le \lambda g\left(y\right)+(1-\lambda )g\left(z\right),\forall \lambda \in [0,1].$$

The function g is referred to as a geodesic strictly convex function at $y\in S$ if the preceding inequality is strict, for $y\ne z$ and $\lambda \in (0,1)$.

In particular, if $\mathcal{H}$ is a Hadamard manifold, then g is referred to as a geodesic convex function at y if, and only if, the following holds:

$$g\left({exp}_y\left(\lambda {exp}_y^{-1}\left(z\right)\right)\right)\le \lambda g\left(y\right)+(1-\lambda )g\left(z\right),\forall z\in S,\forall \lambda \in [0,1].$$

The function g is referred to as a geodesic strictly convex function at $y\in S$ if the preceding inequality is strict, for $y\ne z$ and $\lambda \in (0,1)$.

For more details on geodesic convex functions on Hadamard manifolds, we refer the reader to [5,6,47,48] and the references cited therein. Unless specified otherwise, throughout the remainder of the paper, we shall use the symbol $\mathcal{H}$ to denote a Hadamard manifold of dimension n.

3. Necessary Optimality Conditions for (MMPEC)

In this section, we formulate a multiobjective mathematical programming problem with equilibrium constraints on Hadamard manifolds (MMPEC). We introduce the generalized Guignard constraint qualification for MMPEC, and employ it to deduce the KKT-type necessary optimality criteria for MMPEC.

Let us consider the following multiobjective programming problem with equilibrium constraints on Hadamard manifolds (MMPEC):

$$\begin{array}{cc}\hfill \left(\mathrm{MMPEC}\right)\mathrm{Minimize}f\left(y\right)& :=(f_1\left(y\right),\dots ,f_l\left(y\right)),\hfill \\ \hfill \mathrm{subject}\mathrm{to}g\left(y\right)& \le 0,\hfill \\ \hfill h\left(y\right)& =0,\hfill \\ \hfill P\left(y\right)& \ge 0,\hfill \\ \hfill Q\left(y\right)& \ge 0,\hfill \\ \hfill P{\left(y\right)}^{T}Q\left(y\right)& =0,\hfill \end{array}$$

where $f_j:\mathcal{H}\to \mathbb{R}$, ($j\in \{1,2,\dots ,l\}),g:\mathcal{H}\to {\mathbb{R}}^p,h:\mathcal{H}\to {\mathbb{R}}^q,P:\mathcal{H}\to {\mathbb{R}}^m,Q:$$\mathcal{H}\to {\mathbb{R}}^m$ are all smooth functions that are defined on a Hadamard manifold $\mathcal{H}$. The feasible set of our considered problem (MMPEC) is denoted by the set D.

For any $\tilde{z}\in D$, let us define the following index sets that will be helpful in the subsequent analysis.

$$\begin{array}{cc}\hfill J_g& :=J_g\left(\tilde{z}\right):=\left\{j:g_j\left(\tilde{z}\right)=0\right\},\hfill \\ \hfill \xi & :=\xi \left(\tilde{z}\right):=\left\{j:P_j\left(\tilde{z}\right)=0,Q_j\left(\tilde{z}\right)>0\right\},\hfill \\ \hfill \eta & :=\eta \left(\tilde{z}\right):=\left\{j:P_j\left(\tilde{z}\right)=0,Q_j\left(\tilde{z}\right)=0\right\},\hfill \\ \hfill \zeta & :=\zeta \left(\tilde{z}\right):=\left\{j:P_j\left(\tilde{z}\right)>0,Q_j\left(\tilde{z}\right)=0\right\}.\hfill \end{array}$$

Remark 1.

(a) 

The index sets defined above clearly depend on the feasible element $\tilde{z}\in \mathcal{D}$. However, in the rest of the paper, when such a dependence is easily understandable from the context, we avoid explicitly indicating this.

(b) 

The set η is referred to as the degenerate set. Any arbitrary feasible element $\tilde{z}\in D$ is said to satisfy the strict complementarity condition if $\eta =\varphi$.

We now introduce the function $\theta :\mathcal{H}\to \mathbb{R}$, defined as

$$\theta \left(z\right):=P{\left(z\right)}^{T}Q\left(z\right).$$

Now, we recall the notions of an efficient solution and weakly efficient solution to MMPEC (see, for instance, [40]).

Definition 3.

Any feasible element $\tilde{z}\in D$ is referred to as a Pareto efficient solution of the problem (MMPEC), if there is no element $z\in D$, such that

$$\begin{array}{cc}\hfill f\left(z\right)& ⪯f\left(\tilde{z}\right).\hfill \end{array}$$

Definition 4.

Any feasible element $\tilde{z}\in D$ is referred to as a weak Pareto efficient solution of the problem (MMPEC), if there is no element $z\in D$, such that

$$f\left(z\right)\prec f\left(\tilde{z}\right).$$

Let $\tilde{z}\in D$ be an arbitrary feasible point. Now, we define the sets $S^k$ and S for every $k=1,\dots ,l$ that will be used in the rest of the paper.

$$\begin{array}{cc}\hfill S^k:=\{z\in \mathcal{H}\mid f_j\left(z\right)& \leqq f_j\left(\tilde{z}\right),\forall j\in \{1,\dots ,l\}\mathrm{and}j\ne k,\hfill \\ \hfill g_j\left(z\right)& \leqq 0,\forall j\in \{1,2,\dots ,p\},\hfill \\ \hfill h_j\left(z\right)& =0,\forall j\in \{1,2,\dots ,q\},\hfill \end{array}$$

$$\begin{array}{cc}\hfill P_j\left(z\right)& \geqq 0,\forall j\in \{1,2,\dots ,m\},\hfill \\ \hfill Q_j\left(z\right)& \geqq 0,\forall j\in \{1,2,\dots ,m\},\hfill \\ \hfill P{\left(z\right)}^{T}Q\left(z\right)& =0\}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill S:=\{z\in \mathcal{H}\mid f_j\left(z\right)& \leqq f_j\left(\tilde{z}\right),\forall j\in \{1,\dots ,l\},\hfill \\ \hfill g_j\left(z\right)& \leqq 0,\forall j\in \{1,2,\dots ,p\},\hfill \\ \hfill h_j\left(z\right)& =0,\forall j\in \{1,2,\dots ,q\},\hfill \\ \hfill P_j\left(z\right)& \geqq 0,\forall j\in \{1,2,\dots ,m\},\hfill \\ \hfill Q_j\left(z\right)& \geqq 0,\forall j\in \{1,2,\dots ,m\},\hfill \\ \hfill P{\left(z\right)}^{T}Q\left(z\right)& =0\}.\hfill \end{array}$$

The following definition of the Bouligand tangent cone for a subset of a Hadamard manifold reives from Karkhaneei and Mahdavi-Amiri [43].

Definition 5.

Let $S\subseteq \mathcal{H}$ and $p\in cl\left(S\right)$. The Bouligand tangent cone (in other words, contingent cone) of S at p, denoted by $\mathcal{T}(S,p)$, is defined by

$$\mathcal{T}(S,p)=\left\{v\in T_p\mathcal{H}:\exists t_k\downarrow 0,\exists v_k\in T_p\mathcal{H},v_k\to v,\forall k\in \mathbb{N},{\exp }_p\left(t_k{v}_k\right)\in S\right\}.$$

Now, we extend the definition of linearizing cone from Singh and Mishra [28] for MMPEC on Hadamard manifolds.

Definition 6.

For any $\tilde{z}\in D$, the linearizing cone to S at $\tilde{z}$ is defined as follows:

$$\begin{array}{cccc}\hfill T^{Lin}\left(S,\tilde{z}\right)=\{w\in T_{\tilde{z}}\mathcal{H}:& \langle \mathrm{grad}{f}_j\left(\tilde{z}\right),w\rangle \leqq 0\hfill & \hfill & \forall j\in \{1,2,\dots ,l\},\hfill \\ \hfill & \langle \mathrm{grad}{g}_j\left(\tilde{z}\right),w\rangle \leqq 0,\hfill & \hfill & \forall j\in J_g,\hfill \\ \hfill & \langle \mathrm{grad}{h}_j\left(\tilde{z}\right),w\rangle =0,\hfill & \hfill & \forall j\in \{1,2,\dots ,q\},\hfill \\ \hfill & \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w\rangle =0,\hfill & \hfill & \forall j\in \xi ,\hfill \\ \hfill & \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w\rangle =0,\hfill & \hfill & \forall j\in \zeta ,\hfill \\ \hfill & \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w\rangle \geqq 0,\hfill & \hfill & \forall j\in \eta ,\hfill \\ \hfill & \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w\rangle \geqq 0,\hfill & \hfill & \forall j\in \eta \}.\hfill \end{array}$$

Remark 2.

Definition 6 extends Definition 3.1 from Maeda [40] from Euclidean spaces to Hadamard manifolds, and generalizes this to a wider class of optimization problems, namely, MMPEC. Moreover, it follows from the above definition that

$$\mathcal{T}(S,\tilde{z})\subseteq T^{Lin}\left(S,\tilde{z}\right).$$

To define the generalized Guignard constraint qualification for MMPEC, we now extend the definition of modified linearizing cone given by Singh and Mishra [28] from Euclidean space to Hadamard manifold.

Definition 7.

For any $\tilde{z}\in D$, the modified linearizing cone to S at $\tilde{z}$ is defined as follows:

$$\begin{array}{cccc}\hfill T_{MMPEC}^{Lin}\left(S,\tilde{z}\right)=\{w\in T_{\tilde{z}}\mathcal{H}\mid \langle \mathrm{grad}{f}_j\left(\tilde{z}\right),w\rangle & \leqq 0,\hfill & \hfill & \forall j=1,\dots ,l,\hfill \\ \hfill \langle \mathrm{grad}{g}_j\left(\tilde{z}\right),w\rangle & \leqq 0,\hfill & \hfill & \forall j\in J_g,\hfill \\ \hfill \langle \mathrm{grad}{h}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \{1,2,\dots ,q\},\hfill \\ \hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \xi ,\hfill \\ \hfill \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \zeta ,\hfill \end{array}$$

$$\begin{array}{cccc}\hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w\rangle & \geqq 0,\hfill & \hfill & \forall j\in \eta ,\hfill \\ \hfill \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w\rangle & \geqq 0,\hfill & \hfill & \forall j\in \eta ,\hfill \\ \hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w\rangle \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \eta \}.\hfill \end{array}$$

Remark 3.

From Definitions 6 and 7, we have

$$T_{MMPEC}^{Lin}\left(S,\tilde{z}\right)\subseteq T^{Lin}\left(S,\tilde{z}\right).$$

In the following definition, we present the generalized Guignard constraint qualification for (MMPEC) on Hadamard manifolds.

Definition 8.

Let $\tilde{z}\in D$. Then, the generalized Guignard constraint qualification (GGCQ) is said to be satisfied at $\tilde{z}$ if and only if the following holds:

$$T_{MMPEC}^{Lin}\left(S,\tilde{z}\right)\subseteq \bigcap _{i=1}^{l}clco\mathcal{T}\left(S^i,\tilde{z}\right).$$

In the following theorem, we derive a necessary condition of efficiency for MMPEC.

Theorem 2.

Let $\tilde{z}\in D$ be any arbitrary feasible point of MMPEC. Let us further assume that GGCQ is satisfied at $\tilde{z}\in D$. If $\tilde{z}\in D$ is an efficient solution to MMPEC; then, the following system of inequalities

$$\begin{array}{cccc}\hfill \langle \mathrm{grad}{f}_j\left(\tilde{z}\right),w\rangle & \leqq 0,\hfill & \hfill & \forall j\in \{1,\dots ,l\},\hfill \\ \hfill \langle \mathrm{grad}{f}_j\left(\tilde{z}\right),w\rangle & <0,\hfill & \hfill & \mathrm{for}\mathrm{at}\mathrm{least}\mathrm{one}j,\hfill \\ \hfill \langle \mathrm{grad}{g}_j\left(\tilde{z}\right),w\rangle & \leqq 0,\hfill & \hfill & \forall j\in J_g,\hfill \\ \hfill \langle \mathrm{grad}{h}_j\left(\tilde{z}\right),v\rangle & =0,\hfill & \hfill & \forall j\in \{1,2,\dots ,q\},\hfill \\ \hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \xi ,\hfill \\ \hfill \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \zeta ,\hfill \\ \hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w\rangle & \geqq 0,\hfill & \hfill & \forall j\in \eta ,\hfill \\ \hfill \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w\rangle & \geqq 0,\hfill & \hfill & \forall j\in \eta ,\hfill \\ \hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w\rangle \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \eta ,\hfill \end{array}$$

has no solution $w\in T_{\tilde{z}}\mathcal{H}$.

Proof. 

On the contrary, let us assume that some vector $w\in T_{\tilde{z}}\mathcal{H}$ exists, such that the following system of inequalities is satisfied:

$$\begin{array}{cccc}\hfill \langle \mathrm{grad}{f}_j\left(\tilde{z}\right),w\rangle & \leqq 0,\hfill & \hfill & \forall j\in \{1,\dots ,l\},\hfill \\ \hfill \langle \mathrm{grad}{f}_j\left(\tilde{z}\right),w\rangle & <0,\hfill & \hfill & \mathrm{for}\mathrm{at}\mathrm{least}\mathrm{one}j,\hfill \\ \hfill \langle \mathrm{grad}{g}_j\left(\tilde{z}\right),w\rangle & \leqq 0,\hfill & \hfill & \forall j\in J_g,\hfill \\ \hfill \langle \mathrm{grad}{h}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \{1,2,\dots ,q\},\hfill \\ \hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \xi ,\hfill \\ \hfill \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \zeta ,\hfill \\ \hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w\rangle & \geqq 0,\hfill & \hfill & \forall j\in \eta ,\hfill \\ \hfill \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w\rangle & \geqq 0,\hfill & \hfill & \forall j\in \eta ,\hfill \\ \hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w\rangle \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \eta .\hfill \end{array}$$

Then, from Definition 7, we have

$$w\in T_{MMPEC}^{Lin}\left(S,\tilde{z}\right).$$

Without loss of generality, let us consider that

$$\begin{array}{cc}& 〈grad{f}_1\left(\tilde{z}\right),w〉<0,\hfill \\ & 〈grad{f}_j\left(\tilde{z}\right),w〉\leqq 0,j=2,3,\dots ,l.\hfill \end{array}$$

Since (GGCQ) is satisfied at $\tilde{z}\in D$, it follows that

$$w\in \mathrm{cl}co\mathcal{T}\left(S^1;\tilde{z}\right).$$

Then, it follows that there some sequence ${\left\{w_m\right\}}_{m=1}^{\infty }\subseteq co\mathcal{T}\left(S^1;\tilde{z}\right)$ exists, such that

$$\underset{m\to \infty }{lim}{w}_m=w.$$

For every $m\in \mathbb{N}$, $w_m\in {\left\{w_m\right\}}_{m=1}^{\infty }$. Then, for any element $w_m$ of the sequence ${\left\{w_m\right\}}_{m=1}^{\infty }$, there exists $L_m\in \mathbb{R},{\mu }_{mk}\geqq 0$ and $w_{mk}\in \mathcal{T}\left(S^1,\tilde{z}\right),k=1,2,\dots ,L_m$, such that

$$\sum _{k=1}^{L_m}{\mu }_{mk}=1,\sum _{k=1}^{L_m}{\mu }_{mk}{w}_{mk}=w_m.$$

since $w_{mk}\in \mathcal{T}\left(S^1,\tilde{z}\right)$ for every $m=1,2,\dots$ and $k=1,2,\dots ,L_m$, then by Definition 5, it follows that there are sequences ${\left\{w_{mk}^r\right\}}_{r=1}^{\infty }$, $w_{mk}^r\in T_{\tilde{z}}\mathcal{H}$$\forall r\in \mathbb{N}$ and ${\left\{t_{mk}^r\right\}}_{r=1}^{\infty }$, $t_{mk}^r\left(>0\right)$$\in \mathbb{R}$$\forall r\in \mathbb{N}$, with $t_{mk}^r\downarrow 0$, such that

$$\underset{r\to \infty }{lim}{w}_{mk}^r=w_{mk},{exp}_{\tilde{z}}\left(t_{mk}^r{w}_{mk}^r\right)\in S^1.$$

Let us set $x_{mk}^r$ as follows:

$$x_{mk}^r:={exp}_{\tilde{z}}\left(t_{mk}^r{w}_{mk}^r\right),\forall r\in \mathbb{N}.$$

Then, for every $n\in \mathbb{N}$, we have the following:

$$\begin{array}{cc}\hfill f_j\left(x_{mk}^r\right)& =f_j\left({exp}_{\tilde{z}}\left(t_{mk}^r{w}_{mk}^r\right)\right)\leqq f_j\left(\tilde{z}\right),\forall j\in \{2,3,\dots ,l\},\hfill \\ \hfill g_j\left(x_{mk}^r\right)& =g_j\left({exp}_{\tilde{z}}\left(t_{mk}^r{w}_{mk}^r\right)\right)\leqq 0,\forall j\in J_g,\hfill \\ \hfill h_j\left(x_{mk}^r\right)& =h_j\left({exp}_{\tilde{z}}\left(t_{mk}^r{w}_{mk}^r\right)\right)\leqq 0,\forall j\in \{1,2,\dots ,q\},\hfill \\ \hfill P_j\left(x_{mk}^r\right)& =P_j\left({exp}_{\tilde{z}}\left(t_{mk}^r{w}_{mk}^r\right)\right)=0,\forall j\in \xi ,\hfill \\ \hfill Q_j\left(x_{mk}^r\right)& =Q_j\left({exp}_{\tilde{z}}\left(t_{mk}^r{w}_{mk}^r\right)\right)=0,\forall j\in \zeta ,\hfill \\ \hfill P_j\left(x_{mk}^r\right)& =P_j\left({exp}_{\tilde{z}}\left(t_{mk}^n{w}_{mk}^r\right)\right)\geqq 0,\forall j\in \eta ,\hfill \\ \hfill Q_j\left(x_{mk}^r\right)& =Q_j\left({exp}_{\tilde{z}}\left(t_{mk}^r{w}_{mk}^r\right)\right)\geqq 0,\forall j\in \eta .\hfill \end{array}$$

(1)

Again, since $\tilde{z}\in D$ is an efficient solution of MMPEC, it follows that:

$$f_1\left(x_{mk}^r\right)=f_1\left({exp}_{\tilde{z}}\left(t_{mk}^r{w}_{mk}^r\right)\right)\geqq f_1\left(\tilde{z}\right),\forall r\in \mathbb{N}.$$

From the Taylor expansion of $f_j$ at $\tilde{z}$, for each $j=2,3,\dots ,l$, we have the following

$$f_j\left({exp}_{\tilde{z}}\left(t_{mk}^r{w}_{mk}^r\right)\right)=f_j\left(\tilde{z}\right)+t_{mk}^r\left.〈grad{f}_j\left(\tilde{z}\right)\right),w_{mk}^r〉+o\left(t_{mk}^r\right).$$

Then, it follows that for each $j=2,3,\dots ,l$, we have

$$\frac{f_j\left({exp}_{\tilde{z}}\left(t_{mk}^r{w}_{mk}^r\right)\right)-f_j\left(\tilde{z}\right)}{t_{mk}^r}=〈grad{f}_j\left(\tilde{z}\right),w_{mk}^r〉+\frac{o\left(t_{mk}^r\right)}{t_{mk}^r}.$$

(2)

We have $f_j\left(x_{mk}^r\right)=f_j\left({exp}_{\tilde{z}}\left(t_{mk}^r{w}_{mk}^r\right)\right)\leqq f_j\left(\tilde{z}\right),j=2,3,\dots ,l.$ Then, as $t_{mk}^r\to 0$, it follows from Equation (2) that

$$〈grad{f}_j\left(\tilde{z}\right),w_{mk}〉\leqq 0,j=2,3,\dots ,l.$$

Similarly, we can show that

$$\begin{array}{cc}\hfill \langle \mathrm{grad}{f}_1\left(\tilde{z}\right),w_{mk}\rangle & \geqq 0,\hfill \\ \hfill \langle \mathrm{grad}{f}_j\left(\tilde{z}\right),w_{mk}\rangle & \leqq 0,\hfill & \hfill & j\in \{2,3,\dots ,l\},\hfill \\ \hfill \langle \mathrm{grad}{g}_j\left(\tilde{z}\right),w_{mk}\rangle & \leqq 0,\hfill & \hfill & \forall j\in J_g,\hfill \\ \hfill \langle \mathrm{grad}{h}_j\left(\tilde{z}\right),w_{mk}\rangle & =0,\hfill & \hfill & \forall j\in \{1,2,\dots ,q\},\hfill \\ \hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w_{mk}\rangle & =0,\hfill & \hfill & \forall j\in \xi ,\hfill \\ \hfill \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w_{mk}\rangle & =0,\hfill & \hfill & \forall j\in \zeta ,\hfill \\ \hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w_{mk}\rangle & \geqq 0,\hfill & \hfill & \forall j\in \eta ,\hfill \\ \hfill \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w_{mk}\rangle & \geqq 0,\hfill & \hfill & \forall j\in \eta ,\hfill \\ \hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w_{mk}\rangle \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w_{mk}\rangle & =0,\hfill & \hfill & \forall j\in \eta .\hfill \end{array}$$

From the continuity and linearity property of the inner product, it follows that

$$\begin{array}{cc}\hfill \langle \mathrm{grad}{f}_1\left(\tilde{z}\right),w\rangle & \geqq 0,\hfill \\ \hfill \langle \mathrm{grad}{f}_j\left(\tilde{z}\right),w\rangle & \leqq 0,\hfill & \hfill & j\in \{2,3,\dots ,l\},\hfill \\ \hfill \langle \mathrm{grad}{g}_j\left(\tilde{z}\right),w\rangle & \leqq 0,\hfill & \hfill & \forall j\in J_g,\hfill \\ \hfill \langle \mathrm{grad}{h}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \{1,2,\dots ,q\},\hfill \\ \hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \xi ,\hfill \\ \hfill \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \zeta ,\hfill \\ \hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w\rangle & \geqq 0,\hfill & \hfill & \forall j\in \eta ,\hfill \\ \hfill \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w\rangle & \geqq 0,\hfill & \hfill & \forall j\in \eta ,\hfill \\ \hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w\rangle \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \eta ,\hfill \end{array}$$

which is a contradiction. This completes the proof. □

In the following theorem, we derive the necessary optimality conditions for MMPEC using a generalized Guignard constraint qualification (GGCQ).

Theorem 3.

Let $\tilde{z}\in D$ be an arbitrary feasible point of MMPEC. Let us further assume that GGCQ is satisfied at $\tilde{z}\in D$. If $\tilde{z}\in D$ is an efficient solution of (MMPEC), then there exist Lagrange multipliers ${\tau }_j\in \mathbb{R}$$(j=1,\dots ,l)$, ${\mu }_j^g\in \mathbb{R}$$(j=1,\dots ,p)$, ${\sigma }_j^{+}\in \mathbb{R}$, ${\sigma }_j^{-}\in \mathbb{R}$$(j\in \{1,\dots ,q\left\}\right)$, ${\mu }_j\in \mathbb{R}$$(j\in \{1,\dots ,m\}),{\nu }_j\in \mathbb{R}$$(j\in \{1,\dots ,m\left\}\right),\delta \in \mathbb{R}$, such that

$$\begin{array}{c}\sum _{j=1}^l{\tau }_j\mathrm{grad}{f}_j\left(\tilde{z}\right)+\sum _{j=1}^p{\mu }_j^g\mathrm{grad}{g}_j\left(\tilde{z}\right)+\sum _{j=1}^q{\sigma }_j^{+}\mathrm{grad}{h}_j\left(\tilde{z}\right)\hfill \\ \hfill -\sum _{j=1}^q{\sigma }_j^{-}\mathrm{grad}{h}_j\left(\tilde{z}\right)-\sum _{j=1}^m{\mu }_j\mathrm{grad}{P}_j\left(\tilde{z}\right)-\sum _{j=1}^m{\nu }_j\mathrm{grad}{Q}_j\left(\tilde{z}\right)+\delta \mathrm{grad}\theta \left(\tilde{z}\right)=0\end{array}$$

and

$$\begin{array}{cc}\hfill & {\tau }_j>0,\forall j\in \{1,\dots l\},\hfill \\ \hfill & g_j\left(\tilde{z}\right)\leqq 0,{\mu }_j^g\geqq 0,{\mu }_j^g{g}_j\left(\tilde{z}\right)=0,\forall j\in \{1,\dots ,p\},\hfill \\ \hfill & h_j\left(\tilde{z}\right)\leqq 0,{\sigma }_j^{+}\geqq 0,{\sigma }_j^{+}{h}_j\left(\tilde{z}\right)=0,\forall j\in \{1,\dots ,q\},\hfill \\ \hfill -& h_j\left(\tilde{z}\right)\leqq 0,{\sigma }_j^{-}\geqq 0,{\sigma }_j^{-}\left(-h_j\left(\tilde{z}\right)\right)=0,\forall j\in \{1,\dots ,q\},\hfill \\ \hfill -& P_j\left(\tilde{z}\right)\leqq 0,{\mu }_j\geqq 0,{\mu }_j\left(-P_j\left(\tilde{z}\right)\right)=0,\forall j\in \{1,\dots ,m\},\hfill \\ \hfill -& Q_j\left(\tilde{z}\right)\leqq 0,{\nu }_j\geqq 0,{\nu }_j\left(-Q_j\left(\tilde{z}\right)\right)=0,\forall j\in \{1,\dots ,m\},\hfill \\ \hfill & \theta \left(\tilde{z}\right)\leqq 0,\delta \geqq 0,\delta \theta \left(\tilde{z}\right)=0.\hfill \end{array}$$

Proof. 

Since $\tilde{z}\in D$ is an efficient solution of MMPEC and GGCQ holds at $\tilde{z}$, it follows from Theorem 2 that the following system of inequalities

$$\begin{array}{cccc}\hfill \langle \mathrm{grad}{f}_j\left(\tilde{z}\right),w\rangle & \leqq 0,\hfill & \hfill & \forall j\in \{1,\dots ,l\},\hfill \\ \hfill \langle \mathrm{grad}{f}_j\left(\tilde{z}\right),w\rangle & <0,\hfill & \hfill & \mathrm{for}\mathrm{at}\mathrm{least}\mathrm{one}j,\hfill \\ \hfill \langle \mathrm{grad}{g}_j\left(\tilde{z}\right),w\rangle & \leqq 0,\hfill & \hfill & \forall j\in J_g,\hfill \\ \hfill \langle \mathrm{grad}{h}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \{1,\dots ,q\},\hfill \\ \hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w\rangle & \geqq 0,\hfill & \hfill & \forall j\in \xi \cup \eta ,\hfill \\ \hfill \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w\rangle & \geqq 0,\hfill & \hfill & \forall j\in \zeta \cup \eta ,\hfill \\ \hfill \langle \mathrm{grad}{P}_j\left(\tilde{z}\right),w\rangle \langle \mathrm{grad}{Q}_j\left(\tilde{z}\right),w\rangle & =0,\hfill & \hfill & \forall j\in \eta \hfill \end{array}$$

has no solution $w\in T_{\tilde{z}}\mathcal{H}$. It is well-known that $T_{\tilde{z}}\mathcal{H}$ is a vector space of dimension n. In view of this fact, by employing Tucker’s theorem of alternative ([49] pp. 29–30), it follows that ${\tau }_j\in \mathbb{R}$$(j\in \{1,\dots ,l\left\}\right)$, ${\mu }_j^g\in \mathbb{R}$$(j\in \{1,\dots ,p\left\}\right)$, ${\sigma }_j^{+}$, ${\sigma }_j^{-}\in \mathbb{R}$$(j\in \{1,\dots ,q\left\}\right)$, ${\mu }_j\in \mathbb{R}(j\in \{1,\dots ,m\}),{\nu }_j\in \mathbb{R}(j\in \{1,\dots ,m\}),\delta \in \mathbb{R}$, such that

$$\begin{array}{c}\sum _{j=1}^l{\tau }_{j}grad{f}_j\left(\tilde{z}\right)+\sum _{j\in J_g}{\mu }_j^{g}grad{g}_j\left(\tilde{z}\right)+\sum _{j=1}^q{\sigma }_j^{+}grad{h}_j\left(\tilde{z}\right)-\sum _{j=1}^q{\sigma }_j^{-}grad{h}_j\left(\tilde{z}\right)\hfill \\ \hfill -\sum _{j\in \xi \cup \eta }{\mu }_{j}grad{P}_j\left(\tilde{z}\right)-\sum _{j\in \zeta \cup \eta }{\nu }_{j}grad{Q}_j\left(\tilde{z}\right)+\delta grad\theta \left(\tilde{z}\right)=0,\end{array}$$

By setting ${\mu }_j^g=0$$(\forall j\notin J_g$),${\mu }_j=0$$(\forall j\notin \xi \cup \eta$),${\nu }_j=0$$(\forall j\notin \zeta \cup \eta$), we have

$$\begin{array}{c}\sum _{j=1}^l{\tau }_j\mathrm{grad}{f}_j\left(\tilde{z}\right)+\sum _{j=1}^p{\mu }_j^g\mathrm{grad}{g}_j\left(\tilde{z}\right)+\sum _{j=1}^q{\sigma }_j^{+}\mathrm{grad}{h}_j\left(\tilde{z}\right)\hfill \\ \hfill -\sum _{j=1}^q{\sigma }_j^{-}\mathrm{grad}{h}_j\left(\tilde{z}\right)-\sum _{j=1}^m{\mu }_j\mathrm{grad}{P}_j\left(\tilde{z}\right)-\sum _{j=1}^m{\nu }_j\mathrm{grad}{Q}_j\left(\tilde{z}\right)+\delta \mathrm{grad}\theta \left(\tilde{z}\right)=0.\end{array}$$

However, since $g_j\left(\tilde{z}\right)=0$, $(j\in J_g)$, $h_j\left(\tilde{z}\right)=0$, ($j\in \{1,2,\dots ,q\}$) $P_j\left(\tilde{z}\right)=0$, $(j\in \xi \cup \eta )$, $Q_j\left(\tilde{z}\right)=0$, $(j\in \zeta \cup \eta )$, we have

$$\begin{array}{cc}\hfill {\mu }_j^g{g}_j\left(\tilde{z}\right)& =0,j\in \{1,2,\dots ,p\},\hfill \\ \hfill {\sigma }_j^{+}\left(h_j\left(\tilde{z}\right)\right)& =0,j\in \{1,2,\dots ,q\},\hfill \\ \hfill {\sigma }_j^{-}\left(-h_j\left(\tilde{z}\right)\right)& =0,j\in \{1,2,\dots ,q\},\hfill \\ \hfill {\mu }_j{P}_j\left(\tilde{z}\right)& =0,j\in \{1,2,\dots ,m\},\hfill \\ \hfill {\nu }_j{Q}_j\left(\tilde{z}\right)& =0,j\in \{1,2,\dots ,m\}.\hfill \end{array}$$

Hence, the required equations follow. This completes the proof. □

Now, we deduce the strong KKT type necessary optimality condition for MMPEC using a generalized Guignard constraint qualification (GGCQ).

Theorem 4.

Let $\tilde{z}\in D$ be an efficient solution of MMPEC. Further, let us assume that GGCQ is satisfied at $\tilde{z}$. Then, there exist Lagrange multipliers ${\tau }_j\in \mathbb{R}(j\in \{1,\dots ,l\})$, ${\mu }_j^g\in \mathbb{R}$$(j=1,\dots ,p)$, ${\mu }_j^h\in \mathbb{R}(j\in \{1,\dots ,q\})$, ${\mu }_j^P\in \mathbb{R}(j\in \{1,\dots ,m\})$ and ${\mu }_j^Q\in \mathbb{R}$$(j\in \{1,\dots ,m\left\}\right)$, ${\tau }_j>0,\forall j=1,\dots ,l$, such that

$$\begin{array}{cc}\hfill \sum _{j=1}^l{\tau }_j\mathrm{grad}{f}_j\left(\tilde{z}\right)+& \sum _{j=1}^p{\mu }_j^g\mathrm{grad}{g}_j\left(\tilde{z}\right)+\sum _{j=1}^q{\mu }_j^h\mathrm{grad}{h}_j\left(\tilde{z}\right)\hfill \\ \hfill -& \sum _{j=1}^m\left[{\mu }_j^P\mathrm{grad}{P}_j\left(\tilde{z}\right)+{\mu }_j^Q\mathrm{grad}{Q}_j\left(\tilde{z}\right)\right]=0,\hfill \end{array}$$

(3)

$$\begin{array}{c}\hfill g_j\left(\tilde{z}\right)\leqq 0,{\mu }_j^g\geqq 0,{\mu }_j^g{g}_j\left(\tilde{z}\right)=0,\forall j\in \{1,2,\dots ,p\},\\ \hfill h_j\left(\tilde{z}\right)=0,\forall j\in \{1,2,\dots ,q\},\end{array}$$

(4)

$$\begin{array}{cc}\hfill & {\mu }_j^P\mathrm{free},\forall j\in \xi ,{\mu }_j^P\geqq 0,\forall j\in \eta ,{\mu }_j^P=0,\forall j\in \zeta ,\hfill \\ \hfill & {\mu }_j^Q\mathrm{free},\forall j\in \zeta ,{\mu }_j^Q\geqq 0,\forall j\in \eta ,{\mu }_j^Q=0,\forall j\in \xi .\hfill \end{array}$$

(5)

Proof. 

Since $\tilde{z}\in D$ is an efficient solution of the MMPEC, such that GGCQ is satisfied at $\tilde{z}$, then, from Theorem 3, we can infer that there are Lagrange multipliers ${\tau }_j\in \mathbb{R}(j\in \{1,\dots ,l\}),{\mu }_j^g\in \mathbb{R}(j\in \{1,\dots ,p\})$, ${\sigma }_j^{+},{\sigma }_j^{-}\in \mathbb{R}(j\in \{1,\dots ,q\})$, ${\mu }_j\in \mathbb{R}$$(j\in \{1,\dots ,m\}),{\nu }_j\in \mathbb{R}$$(j\in \{1,\dots ,m\left\}\right),\delta \in \mathbb{R}$, such that

$$\begin{array}{cc}\hfill & \sum _{j=1}^l{\tau }_j\mathrm{grad}{f}_j\left(\tilde{z}\right)+\sum _{j=1}^p{\mu }_j^g\mathrm{grad}{g}_j\left(\tilde{z}\right)+\sum _{j=1}^q{\sigma }_j^{+}\mathrm{grad}{h}_j\left(\tilde{z}\right)-\hfill \\ \hfill & \sum _{j=1}^q{\sigma }_j^{-}\mathrm{grad}{h}_j\left(\tilde{z}\right)-\sum _{j=1}^m{\mu }_j\mathrm{grad}{P}_j\left(\tilde{z}\right)-\sum _{j=1}^m{\nu }_j\mathrm{grad}{Q}_j\left(\tilde{z}\right)+\delta \mathrm{grad}\theta \left(\tilde{z}\right)=0\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & {\tau }_j>0,\forall j\in \{1,2,\dots ,l\},\hfill \\ \hfill & g_j\left(\tilde{z}\right)\leqq 0,{\mu }_j^g\geqq 0,{\mu }_j^g{g}_j\left(\tilde{z}\right)=0,\forall j\in \{1,2,\dots ,p\},\hfill \\ \hfill & h_j\left(\tilde{z}\right)\leqq 0,{\sigma }_j^{+}\geqq 0,{\sigma }_j^{+}{h}_j\left(\tilde{z}\right)=0,\forall j\in \{1,2,\dots ,q\},\hfill \\ \hfill -& h_j\left(\tilde{z}\right)\leqq 0,{\sigma }_j^{-}\geqq 0,{\sigma }_j^{-}\left(-h_j\left(\tilde{z}\right)\right)=0,\forall j\in \{1,2,\dots ,q\},\hfill \\ \hfill -& P_j\left(\tilde{z}\right)\leqq 0,{\mu }_j\geqq 0,{\mu }_j\left(-P_j\left(\tilde{z}\right)\right)=0,\forall j\in \{1,2,\dots ,m\},\hfill \\ \hfill -& Q_j\left(\tilde{z}\right)\leqq 0,{\mu }_j\geqq 0,{\mu }_j\left(-Q_j\left(\tilde{z}\right)\right)=0,\forall j\in \{1,2,\dots ,m\},\hfill \\ \hfill & \theta \left(\tilde{z}\right)\leqq 0,\delta \geqq 0,\delta \theta \left(\tilde{z}\right)=0.\hfill \end{array}$$

Let us set the following

$$\begin{array}{cc}\hfill {\sigma }_j^{+}-{\sigma }_j^{-}& ={\mu }_j^h,\forall j\in \{1,2,\dots ,q\},\hfill \\ \hfill {\mu }_j-\delta Q_j\left(\tilde{z}\right)& ={\mu }_j^P,\forall j\in \{1,2,\dots ,m\},\hfill \\ \hfill {\nu }_j-\delta P_j\left(\tilde{z}\right)& ={\mu }_j^Q,\forall j\in \{1,2,\dots ,m\}.\hfill \end{array}$$

Then, it follows that

$$\begin{array}{cc}\hfill \sum _{j=1}^l{\tau }_j\mathrm{grad}{f}_j\left(\tilde{z}\right)+& \sum _{j=1}^p{\mu }_j^g\mathrm{grad}{g}_j\left(\tilde{z}\right)+\sum _{j=1}^q{\mu }_j^h\mathrm{grad}{h}_j\left(\tilde{z}\right)\hfill \\ \hfill -& \sum _{j=1}^m\left[{\mu }_j^P\mathrm{grad}{P}_j\left(\tilde{z}\right)+{\mu }_j^Q\mathrm{grad}{Q}_j\left(\tilde{z}\right)\right]=0,\hfill \end{array}$$

(6)

$$\begin{array}{c}\hfill g_j\left(\tilde{z}\right)\leqq 0,{\mu }_j^g\geqq 0,{\mu }_j^g{g}_j\left(\tilde{z}\right)=0,\forall j\in \{1,2,\dots ,p\},\\ \hfill h_j\left(\tilde{z}\right)=0,\forall j\in \{1,2,\dots ,q\},\end{array}$$

(7)

$$\begin{array}{cc}\hfill & {\mu }_j^P\mathrm{free},\forall j\in \xi ,{\mu }_j^P\geqq 0,\forall j\in \eta ,{\mu }_j^P=0,\forall j\in \zeta ,\hfill \\ \hfill & {\mu }_j^Q\mathrm{free},\forall j\in \zeta ,{\mu }_j^Q\geqq 0,\forall j\in \eta ,{\mu }_j^Q=0,\forall j\in \xi .\hfill \end{array}$$

(8)

Therefore, the proof is completed. □

Remark 4.

(i) 

The necessary optimality theorem (Theorem 4) extends Theorem 1 of Singh and Mishra [28] on a more general space, namely, Hadamard manifold;

(ii) 

The necessary optimality theorem (Theorem 4) extends Theorem 3.2 of Maeda [40] to a more general space, namely, Hadamard manifold, and also generalizes this for multiobjective programming problems with equilibrium constraints.

The following example illustrates the significance of Theorem 4.

Example 1.

Let us consider the set $\mathcal{H}\subset {\mathbb{R}}^2$ as $\mathcal{H}:=\{x=(x_1,x_2)\in {\mathbb{R}}^2,x_1,x_2>0\}$. Then, $\mathcal{H}$ is a Riemannian manifold (see, for instance, [5,50], and Example 4.4 of [51]). $\mathcal{H}$ is equipped with the metric ${\langle p,q\rangle }_x=\langle \mathcal{G}\left(x\right)p,q\rangle ,\forall p,q\in T_x\mathcal{H}={\mathbb{R}}^2$, where $\langle ·,·\rangle$ denotes the standard inner product on ${\mathbb{R}}^2$ and

$$\mathcal{G}\left(y\right)=\left(\begin{array}{cc}\frac{1}{y_1^2}& 0\\ 0& \frac{1}{y_2^2}\end{array}\right).$$

$\mathcal{H}$ is also a Hadamard manifold. The exponential map ${\exp }_x:T_x\mathcal{H}\to \mathcal{H}$ for any $u\in T_x\mathcal{H}$ is given by ${\exp }_x\left(u\right)=(x_1{e}^{\frac{u_1}{x_1}},x_2{e}^{\frac{u_2}{x_2}})$, $\forall u=(u_1,u_2)\in \mathcal{H}$. The inverse of the exponential map ${\exp }_x^{-1}:\mathcal{H}\to T_x\mathcal{H}$ for any $y\in \mathcal{H}$ is given by ${\exp }_x^{-1}\left(y\right)=\left(x_{1}ln\frac{y_1}{x_1},x_{2}ln\frac{y_2}{x_2}\right)$.

We consider the following multiobjective programming problem with equilibrium constraints on $\mathcal{H}$

$$\begin{array}{cc}\hfill \left(P\right)\mathrm{Minimize}f\left(x\right)& :=(\sqrt{x_1},log{x}_2),\hfill \\ \hfill \mathrm{subject}\mathrm{to}g\left(x\right)& :=1-ln{x}_1-ln{x}_2\le 0,\hfill \\ \hfill P\left(x\right)& :=ln{x}_1-1\ge 0,\hfill \\ \hfill Q\left(x\right)& :=ln{x}_2-1\ge 0,\hfill \\ \hfill P{\left(x\right)}^{T}Q\left(x\right)& :=(ln{x}_1-1)(ln{x}_2-1)=0,\hfill \end{array}$$

where $f_i:\mathcal{H}\to \mathbb{R},(i=1,2),g:\mathcal{H}\to \mathbb{R},P:\mathcal{H}\to \mathbb{R},Q:\mathcal{H}\to \mathbb{R}$ are smooth, real valued functions, defined on $\mathcal{H}$. The feasible set F for the problem is:

$$F=\left\{x\in \mathcal{H},x_1=e,x_2\ge e,\mathrm{or},x_1\ge e,x_2=e\right\}.$$

Let us consider the feasible point $\tilde{z}=(e,e)\in F$. Then, it can be verified that (GGCQ) is satisfied at $\tilde{z}$. Further, it can be verified that $\tilde{z}=(e,e)$ is an efficient solution to the problem $\left(P\right)$.

Now, we have the following

$$\begin{array}{cc}\hfill grad{f}_1\left(x\right)& =\mathcal{G}{\left(x\right)}^{-1}\left(\begin{array}{c}\frac{1}{2\sqrt{x_1}}\\ 0\end{array}\right)=\left(\begin{array}{c}\frac{1}{2}{x}_1\sqrt{x_1}\\ 0\end{array}\right),\hfill \\ \hfill grad{f}_2\left(x\right)& =\mathcal{G}{\left(x\right)}^{-1}\left(\begin{array}{c}0\\ \frac{1}{x_2}\end{array}\right)=\left(\begin{array}{c}0\\ x_2\end{array}\right),\hfill \\ \hfill gradg\left(x\right)& =\mathcal{G}{\left(x\right)}^{-1}\left(\begin{array}{c}-\frac{1}{x_1}\\ -\frac{1}{x_2}\end{array}\right)=\left(\begin{array}{c}-x_1\\ -x_2\end{array}\right),\hfill \\ \hfill gradP\left(x\right)& =\mathcal{G}{\left(x\right)}^{-1}\left(\begin{array}{c}\frac{1}{x_1}\\ 0\end{array}\right)=\left(\begin{array}{c}{x}_1\\ 0\end{array}\right),\hfill \\ \hfill gradQ\left(x\right)& =\mathcal{G}{\left(x\right)}^{-1}\left(\begin{array}{c}0\\ \frac{1}{x_2}\end{array}\right)=\left(\begin{array}{c}0\\ x_2\end{array}\right).\hfill \end{array}$$

Substituting $\tilde{z}=(e,e)$ for $x=(x_1,x_2)$ in the above expressions, we obtain the following

$$\begin{array}{cc}\hfill grad{f}_1\left(\tilde{z}\right)& ={\left(\frac{e\sqrt{e}}{2},0\right)}^T,\hfill \\ \hfill grad{f}_2\left(\tilde{z}\right)& ={\left(0,e\right)}^T,\hfill \\ \hfill gradg\left(\tilde{z}\right)& ={\left(-e,-e\right)}^T,\hfill \\ \hfill gradP\left(\tilde{z}\right)& ={\left(e,0\right)}^T,\hfill \\ \hfill gradQ\left(\tilde{z}\right)& ={\left(0,e\right)}^T.\hfill \end{array}$$

Then, by choosing multipliers as ${\tau }_1=\frac{1}{2},{\tau }_2=\frac{1}{2},{\mu }^g=0,{\mu }^P=\frac{\sqrt{e}}{4},{\mu }^Q=\frac{1}{2}$, we obtain the following

$$\sum _{j=1}^2{\tau }_j\mathrm{grad}{f}_j\left(\tilde{z}\right)+{\mu }^{g}gradg\left(\tilde{z}\right)-\left[{\mu }^{P}gradP\left(\tilde{z}\right)+{\mu }^Q\mathrm{grad}Q\left(\tilde{z}\right)\right]=(0,0).$$

Thus, the necessary optimality condition (Theorem 4) for the problem (P) is verified.

4. Sufficient Optimality Conditions for MMPEC

In this section, we derive sufficient optimality conditions for (MMPEC) under geodesic convexity assumptions.

The following definition extends the definition of M-stationary point (Definition 2.6) given by Ye [35] from Euclidean space to Hadamard manifolds for MMPEC.

Definition 9.

(M-stationary point) Let $\tilde{z}\in D$. Then $\tilde{z}$ is said to be a Mordukhovich stationary point (M-stationary point) if ${\tau }_j>0,$$(j\in \{1,\dots ,l\left\}\right)$ and $\mu =\left({\mu }^g,{\mu }^h,{\mu }^P,{\mu }^Q\right)\in$${\mathbb{R}}^{p+q+2m}$, such that the following conditions are satisfied:

$$\begin{array}{cc}\hfill \sum _{j=1}^l{\tau }_j\mathrm{grad}{f}_j\left(\tilde{z}\right)+& \sum _{j\in J_g}{\mu }_j^g\mathrm{grad}{g}_j\left(\tilde{z}\right)+\sum _{j=1}^q{\mu }_j^h\mathrm{grad}{h}_j\left(\tilde{z}\right)\hfill \\ \hfill -& \sum _{j=1}^m\left[{\mu }_j^P\mathrm{grad}{P}_j\left(\tilde{z}\right)+{\mu }_j^Q\mathrm{grad}{Q}_j\left(\tilde{z}\right)\right]=0,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & {\mu }_j^g\ge 0,\forall j\in J_g,\hfill \\ \hfill & {\mu }_j^P=0,\forall j\in \zeta ,\hfill \\ \hfill & {\mu }_j^Q=0,\forall j\in \xi ,\hfill \\ \hfill & \forall j\in \eta ,\mathrm{either},{\mu }_j{}^P>0,{\mu }_j^Q>0\mathrm{or},{\mu }_j^P{\mu }_j^Q=0.\hfill \end{array}$$

(10)

The following index sets, as defined below, will be helpful to derive the sufficient optimality condition for MMPEC on Hadamard manifold.

$$\begin{array}{cc}& {\eta }^{+}:=\left\{j\in \eta :{\mu }_j^P>0,{\mu }_j^Q>0\right\},\hfill \\ & {\eta }_P^{+}:=\left\{j\in \eta :{\mu }_j^P=0,{\mu }_j^Q>0\right\},\hfill \\ & {\eta }_P^{-}:=\left\{j\in \eta :{\mu }_j{}^P=0,{\mu }_j^Q<0\right\},\hfill \\ & {\eta }_Q^{+}:=\left\{j\in \eta :{\mu }_j^Q=0,{\mu }_j^P>0\right\},\hfill \\ & {\eta }_Q^{-}:=\left\{j\in \eta :{\mu }_j^Q=0,{\mu }_j^P<0\right\},\hfill \\ & {\xi }^{+}:=\left\{j\in \xi :{\mu }_j^P>0\right\},\hfill \end{array}$$

$$\begin{array}{cc}& {\xi }^{-}:=\left\{j\in \xi :{\mu }_j^P<0\right\},\hfill \\ & {\zeta }^{+}:=\left\{j\in \zeta :{\mu }_j^Q>0\right\},\hfill \\ & {\zeta }^{-}:=\left\{j\in \zeta :{\mu }_j^Q<0\right\},\hfill \\ & J^{+}:=\left\{i:{\mu }_j^h>0\right\},\hfill \\ & J^{-}:=\left\{i:{\mu }_j^h<0\right\}.\hfill \end{array}$$

Now, we derive sufficient optimality conditions for MMPEC using geodesic convexity assumptions.

Theorem 5.

Let $\tilde{y}\in D$ be a M-stationary point of MMPEC. Let us assume that each $f_j$$(j=1,\dots ,l)$, $g_j$$\left(j\in J_g\right)$, $h_j$$\left(j\in J^{+}\right)$, $-h_j$$\left(j\in J^{-}\right)$, $P_j$$(j\in {\xi }^{-}\cup {\eta }_Q^{-})$, $-P_j$$(j\in {\xi }^{+}\cup {\eta }_Q^{+}$$\cup {\eta }^{+})$, $Q_j$$\left(j\in {\zeta }^{-}\cup {\eta }_P^{-}\right)$, $-Q_j$$(j\in {\zeta }^{+}\cup {\eta }_P^{+}\cup {\eta }^{+})$ are geodesic convex functions at $\tilde{y}$. If

$${\xi }^{-}\cup {\zeta }^{-}\cup {\eta }_P^{-}\cup {\eta }_Q^{-}=\varphi ,$$

then $\tilde{y}$ is a weak Pareto efficient solution of MMPEC.

Proof. 

However, let us suppose that $\tilde{y}\in D$ is not a weak Pareto efficient solution of MMPEC. Then, it follows that there exists some $x\in D$, such that

$$f_j\left(x\right)<f_j\left(\tilde{y}\right),\forall j=1,\dots ,l.$$

From the geodesic convexity assumption on $f_j$, $j=1,\dots ,l$, it follows that

$$〈\mathrm{grad}{f}_j\left(\tilde{y}\right),{exp}_{\tilde{y}}^{-1}\left(x\right)〉<0,\forall j=1,\dots ,l.$$

Since $\tilde{y}\in D$ is a M-stationary point, there exist ${\tau }_j>0,$$(\forall j=1,\dots ,l)$ and $\mu =\left({\mu }^g,{\mu }^h,{\mu }^P,{\mu }^Q\right)$∈${\mathbb{R}}^{p+q+2m}$, such that

$$\begin{array}{cc}\hfill & \sum _{j=1}^l{\tau }_j\mathrm{grad}{f}_j\left(\tilde{y}\right)+\sum _{j\in J_g}{\mu }_j^g\mathrm{grad}{g}_j\left(\tilde{y}\right)+\sum _{j=1}^q{\mu }_j^h\mathrm{grad}{h}_j\left(\tilde{y}\right)\hfill \\ \hfill -& \sum _{j=1}^m\left[{\mu }_j^P\mathrm{grad}{P}_j\left(\tilde{y}\right)+{\mu }_j^Q\mathrm{grad}{Q}_j\left(\tilde{y}\right)\right]=0,\hfill \end{array}$$

(11)

where,

$$\begin{array}{cc}\hfill & {\mu }_j^g\ge 0,\forall j\in J_g,\hfill \\ \hfill & {\mu }_j^P=0,\forall j\in \zeta ,\hfill \\ \hfill & {\mu }_j^Q=0,\forall j\in \xi ,\hfill \\ \hfill & \forall j\in \eta ,\mathrm{either},{\mu }_j{}^P>0,{\mu }_j^Q>0\mathrm{or},{\mu }_j^P{\mu }_j^Q=0.\hfill \end{array}$$

(12)

Since ${\tau }_j>0,\forall j=1,\dots ,l$, we have the following

$$〈\sum _{j=1}^l{\tau }_j\mathrm{grad}{f}_j\left(\tilde{y}\right),{exp}_{\tilde{y}}^{-1}\left(x\right)〉<0.$$

(13)

From the given hypothesis, we can see that $\tilde{y}$ is a M-stationary point of (MMPEC). Hence, it follows from (11)–(13) that

$$\begin{array}{cc}\hfill \langle \sum _{j\in J_g}{\mu }_j^g\mathrm{grad}{g}_j\left(\tilde{y}\right)+& \sum _{j=1}^q{\mu }_j^h\mathrm{grad}{h}_j\left(\tilde{y}\right)\hfill \\ \hfill -& \sum _{j=1}^m[{\mu }_j^P\mathrm{grad}{P}_j\left(\tilde{y}\right)+{\mu }_j^Q\mathrm{grad}{Q}_j\left(\tilde{y}\right)],{exp}_{\tilde{y}}^{-1}\left(x\right)\rangle >0.\hfill \end{array}$$

(14)

For every $j\in J_g\left(\tilde{y}\right)$, we have $g_j\left(x\right)\leqq 0=g_j\left(\tilde{y}\right)$. Then, from the geodesic convexity assumption on $g_j$ at $\tilde{y}$, we can obtain the following

$$〈\mathrm{grad}{g}_j\left(\tilde{y}\right),{exp}_{\tilde{y}}^{-1}\left(x\right)〉\leqq 0,\forall j\in J_g.$$

(15)

For every $j\in J^{-}$, we have $h_j\left(x\right)\leqq 0=h_j\left(\tilde{y}\right)$. Then, from the geodesic convexity assumption on $h_j$ at $\tilde{y}$, we can obtain

$$〈\mathrm{grad}{h}_j\left(\tilde{y}\right),{exp}_{\tilde{y}}^{-1}\left(x\right)〉\leqq 0,\forall j\in J^{-}.$$

(16)

Similarly, we can yield the following

$$〈\mathrm{grad}{h}_j\left(\tilde{y}\right),{exp}_{\tilde{y}}^{-1}\left(x\right)〉\geqq 0,\forall j\in J^{+}.$$

(17)

Again, $-P_j\left(x\right)\leqq 0=-P_j\left(\tilde{y}\right),\forall j\in {\xi }^{+}\cup {\eta }_Q^{+}$. Then, from the geodesic convexity assumption on $P_j$ at $\tilde{y}$, we can obtain

$$〈\mathrm{grad}{P}_j\left(\tilde{y}\right),{exp}_{\tilde{y}}^{-1}\left(x\right)〉\geqq 0,\forall j\in {\xi }^{+}\cup {\eta }_Q^{+}.$$

(18)

Using similar arguments, we can obtain the following

$$〈\mathrm{grad}{Q}_j\left(\tilde{y}\right),{exp}_{\tilde{y}}^{-1}\left(x\right)〉\geqq 0,\forall j\in {\zeta }^{+}\cup {\eta }_P^{+}.$$

(19)

Again, from the given hypothesis, we have ${\xi }^{-}\cup {\zeta }^{-}\cup {\eta }_P^{-}\cup {\eta }_Q^{-}=\varphi$. Then, from Equations (15)–(19) and the definitions of the index sets, it follows that

$$\begin{array}{cccc}\hfill & 〈\sum _{j\in J_g}{\mu }_j^g\mathrm{grad}{g}_j\left(\tilde{y}\right),{exp}_{\tilde{y}}^{-1}\left(x\right)〉\hfill & \hfill & \leqq 0,\hfill \\ \hfill & 〈\sum _{j=1}^q{\mu }_j^h\mathrm{grad}{h}_j\left(\tilde{y}\right),{exp}_{\tilde{y}}^{-1}\left(x\right)〉\hfill & \hfill & \leqq 0,\hfill \\ \hfill & 〈\sum _{j\in \xi \cup \eta }{\mu }_j^P\mathrm{grad}{P}_j\left(\tilde{y}\right),{exp}_{\tilde{y}}^{-1}\left(x\right)〉\hfill & \hfill & \geqq 0,\hfill \\ \hfill & 〈\sum _{j\in \eta \cup \zeta }{\mu }_j^Q\mathrm{grad}{Q}_j\left(\tilde{y}\right),{exp}_{\tilde{y}}^{-1}\left(x\right)〉\hfill & \hfill & \geqq 0.\hfill \end{array}$$

(20)

Combining the inequalities in (20), we obtain the following:

$$\begin{array}{cc}\hfill \langle \sum _{j\in J_g}{\mu }_j^g\mathrm{grad}{g}_j\left(\tilde{y}\right)+& \sum _{j=1}^q{\mu }_j^h\mathrm{grad}{h}_j\left(\tilde{y}\right)-\sum _{j=1}^m[{\mu }_j^P\mathrm{grad}{P}_j\left(\tilde{y}\right)\hfill \\ & +{\mu }_j^Q\mathrm{grad}{Q}_j\left(\tilde{y}\right)],{exp}_{\tilde{y}}^{-1}\left(x\right)\rangle \leqq 0,\hfill \end{array}$$

which is a contradiction to (14). Hence, we can conclude that $\tilde{y}\in D$ is a weakly efficient solution of (MMPEC). This completes the proof. □

Remark 5.

The sufficient optimality condition (Theorem 5) extends Theorem 2.3 of Ye [35] on a more general space, namely, Hadamard manifold, and generalizes it to a more general class of programming problems, that is, multiobjective programming problems with equilibrium constraints.

The following example illustrates the significance of Theorem 5.

Example 2.

Let us consider the set $\mathcal{H}\subset {\mathbb{R}}^2$ as $\mathcal{H}:=\{x=(x_1,x_2)\in {\mathbb{R}}^2,x_1,x_2>0\}$. Then, $\mathcal{H}$ is a Riemannian manifold (see, for instance, [5,50], and Example 4.4 of [51]). $\mathcal{H}$ is equipped with the metric ${\langle p,q\rangle }_x=\langle \mathcal{G}\left(x\right)p,q\rangle ,\forall p,q\in T_x\mathcal{H}={\mathbb{R}}^2$, where $\langle ·,·\rangle$ denotes the standard inner product on ${\mathbb{R}}^2$ and

$$\mathcal{G}\left(y\right)=\left(\begin{array}{cc}\frac{1}{y_1^2}& 0\\ 0& \frac{1}{y_2^2}\end{array}\right).$$

$\mathcal{H}$ is also a Hadamard manifold. The exponential map ${\exp }_x:T_x\mathcal{H}\to \mathcal{H}$ for any $u\in T_x\mathcal{H}$ is given by ${\exp }_x\left(u\right)=(x_1{e}^{\frac{u_1}{x_1}},x_2{e}^{\frac{u_2}{x_2}})$, $\forall u=(u_1,u_2)\in \mathcal{H}$. The inverse of the exponential map ${\exp }_x^{-1}:\mathcal{H}\to T_x\mathcal{H}$ for any $y\in \mathcal{H}$ is given by ${\exp }_x^{-1}\left(y\right)=\left(x_{1}ln\frac{y_1}{x_1},x_{2}ln\frac{y_2}{x_2}\right)$.

We consider the multiobjective programming problem with equilibrium constraints (P), as defined in Example 1 on $\mathcal{H}$. The feasible set F for the problem is

$$F=\left\{x\in \mathcal{H},x_1=e,x_2\ge e,\mathrm{or},x_1\ge e,x_2=e\right\}.$$

Let us consider the feasible point $\tilde{z}=(e,e)\in F$. Now, we have the following

$$\begin{array}{cc}\hfill grad{f}_1\left(x\right)& =\mathcal{G}{\left(x\right)}^{-1}\left(\begin{array}{c}\frac{1}{2\sqrt{x_1}}\\ 0\end{array}\right)=\left(\begin{array}{c}\frac{1}{2}{x}_1\sqrt{x_1}\\ 0\end{array}\right),\hfill \\ \hfill grad{f}_2\left(x\right)& =\mathcal{G}{\left(x\right)}^{-1}\left(\begin{array}{c}0\\ \frac{1}{x_2}\end{array}\right)=\left(\begin{array}{c}0\\ x_2\end{array}\right),\hfill \\ \hfill gradg\left(x\right)& =\mathcal{G}{\left(x\right)}^{-1}\left(\begin{array}{c}-\frac{1}{x_1}\\ -\frac{1}{x_2}\end{array}\right)=\left(\begin{array}{c}-x_1\\ -x_2\end{array}\right),\hfill \\ \hfill gradP\left(x\right)& =\mathcal{G}{\left(x\right)}^{-1}\left(\begin{array}{c}\frac{1}{x_1}\\ 0\end{array}\right)=\left(\begin{array}{c}{x}_1\\ 0\end{array}\right),\hfill \\ \hfill gradQ\left(x\right)& =\mathcal{G}{\left(x\right)}^{-1}\left(\begin{array}{c}0\\ \frac{1}{x_2}\end{array}\right)=\left(\begin{array}{c}0\\ x_2\end{array}\right).\hfill \end{array}$$

Substituting $\tilde{z}=(e,e)$ for $x=(x_1,x_2)$ in the above equations, we obtain the following

$$\begin{array}{cc}\hfill grad{f}_1\left(\tilde{z}\right)& =\left(\frac{e\sqrt{e}}{2},0\right),\hfill \\ \hfill grad{f}_2\left(\tilde{z}\right)& =\left(0,e\right),\hfill \\ \hfill gradg\left(\tilde{z}\right)& =\left(-e,-e\right),\hfill \\ \hfill gradP\left(\tilde{z}\right)& =\left(e,0\right),\hfill \\ \hfill gradQ\left(\tilde{z}\right)& =\left(0,e\right),\hfill \end{array}$$

Then, by choosing the following multipliers as ${\tau }_1=\frac{1}{2},{\tau }_2=\frac{1}{2},{\mu }^g=0,{\mu }^P=\frac{\sqrt{e}}{4},{\mu }^Q=\frac{1}{2}$, we can obtain the following

$$\sum _{j=1}^2{\tau }_j\mathrm{grad}{f}_j\left(\tilde{z}\right)+{\mu }^{g}gradg\left(\tilde{z}\right)-\left[{\mu }^{P}gradP\left(\tilde{z}\right)+{\mu }^Q\mathrm{grad}Q\left(\tilde{z}\right)\right]=(0,0).$$

This shows that $\overline{z}=(e,e)$ is an M-stationary point of (P). Further, the functions $f_i\left(x\right),$$(i=1,2),g\left(x\right),P\left(x\right),Q\left(x\right)$ are all geodesic convex functions at $\overline{z}=(e,e)$ and all the conditions of the sufficient optimality theorem (Theorem 5) are satisfied. Then, using Theorem 5, one can verify that $\overline{z}=(e,e)$ is a weakly efficient solution of (P).

5. Applications

In this section, we illustrate the significance of our results in Euclidean spaces and in practical engineering problems.

If $\mathcal{H}={\mathbb{R}}^n$, then the results derived in the paper extend the corresponding results deduced by [35,40] for a more general class of optimization problems, namely, the multiobjective mathematical programming problems with equilibrium constraints (MMPEC). Moreover, if $\mathcal{H}={\mathbb{R}}^n$, then the results derived in this paper reduce to the corresponding results derived by Singh and Mishra [28]. Moreover, the results of this paper generalize some well-known corresponding results in Euclidean spaces; see, for instance, [19,23,41].

A very interesting category of problems that can be modelled in terms of (MPEC) is the design of multistage processes involving chemical equilibrium, as studied by Raghunathan and Biegler [22]. They showed that the phase identification problem can be modelled as a minimization problem of Gibbs free energy. Following the work of Raghunathan and Biegler [22], the design of the chemical equilibrium based processes can be modelled in the following manner:

$$\begin{array}{c}\hfill \left(P1\right)\underset{y_1,y_2}{min}{\varphi }_1\left(y_1,y_2\right),\end{array}$$

subject to

$$\begin{array}{cc}\hfill h_1\left(y_1,y_2\right)& \le 0,\hfill \\ \hfill c_1\left(y_1,y_2\right)& =0.\hfill \end{array}$$

$$\begin{array}{c}\hfill \left(P2\right)\underset{y_2}{min}{\varphi }_2\left(y_1,y_2\right),\end{array}$$

subject to

$$\begin{array}{cc}\hfill h_2\left(y_1,y_2\right)& \le 0,\hfill \\ \hfill c_2\left(y_1,y_2\right)& =0,\hfill \end{array}$$

where ${\varphi }_1:{\mathbb{R}}^{n_1+n_2}\to \mathbb{R}$, $h_1:{\mathbb{R}}^{n_1+n_2}\to {\mathbb{R}}^{n_{k_1}}$, $c_1:{\mathbb{R}}^{n_1+n_2}\to {\mathbb{R}}^{n_1},{\varphi }_2:{\mathbb{R}}^{n_1+n_2}\to \mathbb{R}$, $h_2:{\mathbb{R}}^{n_1+n_2}\to {\mathbb{R}}^{n_{h_2}}$ and $c_2:{\mathbb{R}}^{n_1+n_2}\to {\mathbb{R}}^{n_{c_2}}$. In a chemical-equilibrium-based process design, the objective function corresponding to $\left(P1\right)$ signifies the operating cost. The constraints corresponding to $\left(P1\right)$ signify mass, component, energy balances and bounds on flow variables and temperatures. Problem $\left(P2\right)$ deals with minimizing Gibbs free energy.

The usual procedure to solve these problems involves replacing ($P1$) with its stationary conditions. By reformulating the problem ($P1$) with the corresponding stationary conditions, we can arrive at a mathematical programming problem with equilibrium constraints. For more details, we refer to the paper by Raghunathan and Biegler [22].

There are several advantages of formulating such problems in science and engineering using the framework of manifolds. For instance, a complicated constrained mathematical programming problem can be reduced to a simpler unconstrained programming problem by making suitable use of the Riemannian geometry. Secondly, many non-convex optimization problems can be suitably transformed into a convex optimization problem by endowing the space with an appropriate Riemannian metric. For instance, following the work of Colao et al. [52], we consider the following problem:

$$\underset{z\in \mathcal{B}}{min}\varphi \left(z\right)$$

where the feasible set $\mathcal{B}$ is the positive orthant, defined as follows:

$$\mathcal{B}=\left\{z\in {\mathbb{R}}^n:z_j>0,j=1,\dots ,n\right\}.$$

Colao et al. [52] considered that $\varphi :\mathcal{B}\to \mathbb{R}$ is a nonconvex function, defined as follows:

$$\varphi \left(z_1,\dots ,z_n\right)=\sum _{i=1}^m{\alpha }_i\prod _{j=1}^n{z}_j^{{\beta }_{ij}},$$

where ${\alpha }_i\in \mathcal{B}$ and ${\beta }_{ij}\in \mathbb{R}$ for every $i,j$. That is, for every $z\in \mathcal{B}$, and $p,q\in T_z\mathcal{B}$, we have the following:

$${\langle p,q\rangle }_z=\langle G\left(z\right)p,v\rangle =\sum _{i=1}^n\frac{p_i{q}_i}{z_i^2},$$

where $G\left(z\right)$ is the Dikin metric, as defined by Colao et al. [52]. It is a well-known fact that the feasible set $\mathcal{B}$, as defined above, endowed with the Dikin metric, is a Hadamard manifold. It was further verified in [52] that $\varphi$ is geodesic convex function on the feasible set $\mathcal{B}$, with respect to the Dikin metric. As a result, the non-convex constrained optimization problem can be transformed into a convex unconstrained optimization problem by the appropriate use of the Dikin metric.

In light of the above discussions, it is evident that the results deduced in the paper can be applied to a very wide range of practical problems, which arise in several applications of science and engineering.

6. Conclusions and Future Directions

This paper is concerned with a class of multiobjective programming problems with equilibrium constraints on Hadamard manifolds (MMPEC). We introduced the generalized Guignard constraint qualification (GGCQ) on Hadamard manifold for our considered problem (MMPEC). By employing GGCQ, we derived the Karush–Kuhn–Tucker-type necessary optimality criteria for MMPEC. Further, we derived sufficient optimality criteria for (MMPEC) using geodesic convexity assumptions. An interesting application of mathematical programming problems with equilibrium constraints in the field of process engineering is provided, and the advantages of modelling such problems on manifolds is discussed. Several non-trivial examples have been cited in this paper to illustrate the significance of the derived results.

We have extended several of the well-known known results existing in the literature (for instance, [19,23,28,35,40,41,53]) from Euclidean spaces to a more general space, that is, Hadamard manifolds, and for a more general class of optimization problems, namely, mathematical programming problems with equilibrium constraints. To the best of our knowledge, the generalized Guignard constraint qualification and optimality conditions of (MMPEC) on Hadamard manifolds have been not been studied to date.

The sufficient optimality conditions derived in the paper could be further generalized by employing the notions of geodesic pseudoconvex and geodesic quasiconvex functions. Duality models for (MMPEC) have not been considered in the paper, and could be our future course of action. Another limitation of our work is that the results derived in the paper are restricted to the case when the objective and the constraint functions are smooth. The results presented in the paper can be further improved and extended to non-smooth multiobjective programming problems with equilibrium constraints on Hadamard manifolds by employing Mordukhovich limiting subdifferential, which has a better Lagrange multiplier rule compared to the Clarke subdifferential. We would like to address these in our next work.