Necessary and sufficient optimality conditions for vector equilibrium problems on Hadamard manifolds

symmetry S S Article Necessary and Sufficient Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds Gabriel Ruiz-Garzón 1,*,†,‡ , Rafaela Osuna-Gómez 2,‡ and Jaime Ruiz-Zapatero 3,‡ 1Departamento de Estadística e I.O., Universidad de Cádiz, 11405 Cádiz, Spain 2Departamento de Estadística e I.O., Universidad de Sevilla, 41012 Sevilla, Spain 3Department of Physics and Astronomy, University College of London, London WC1E 6BT, UK *Correspondence: gabriel.r[email protected] † Current address: Departamento de Estadística e I.O., Universidad de Cádiz, Campus de Jerez de la Frontera, Avda. de la Universidad s/n, 11405, Jerez de la Frontera, Cádiz, Spain. ‡ These authors contributed equally to this work. Received: 18 July 2019; Accepted: 8 August 2019; Published: 12 August 2019   Abstract: The aim of this paper is to show the existence and attainability of Karush–Kuhn–Tucker optimality conditions for weakly efficient Pareto points for vector equilibrium problems with the addition of constraints in the novel context of Hadamard manifolds, as opposed to the classical examples of Banach, normed or Hausdorff spaces. More specifically, classical necessary and sufficient conditions for weakly efficient Pareto points to the constrained vector optimization problem are presented. The results described in this article generalize results obtained by Gong (2008) and Wei and Gong (2010) and Feng and Qiu (2014) from Hausdorff topological vector spaces, real normed spaces, and real Banach spaces to Hadamard manifolds, respectively. This is done using a notion of Riemannian symmetric spaces of a noncompact type as special Hadarmard manifolds. Keywords: vector equilibrium problem; generalized convexity; hadamard manifolds; weakly efficient pareto points 1. Introduction The pursuit of equilibrium is a ubiquitous horizon in practically all areas of human activity. For example, in economics, the dynamics of offer and demand are typically described as equilibrium problems. In the same way, physical or social phenomena such as the distribution of particles in a container, traffic flow or telecommunication networks can be accurately conceptualized in terms of equilibrium. However, it was not until Fan [ 1 ] that equilibrium theory was applied in the context of Euclidean spaces. Mathematically, the simplest definition of a equilibrium problem consists in finding x∈S such that F(x,y)≥0, ∀y∈S where S⊆Rp is a nonempty closed set and F:Rp×Rp→R is an equilibrium bifunction, i.e., F(x,x) = 0 for all x∈S. Some of the main mathematical problems that can be phrased as equilibrium problems are: Symmetry 2019,11, 1037; doi:10.3390/sym11081037 www.mdpi.com/journal/symmetry Symmetry 2019,11, 1037 2 of 12 • The weak minimum point of a multiobjective function f= (f1 , . . . , fp) over a closed set S⊆Rp is any ¯ x∈S such that for any y∈S , ∃i such that fi(y)−fi(¯ x)≥ 0. Finding a weak minimum point can be reduced to solving an equilibrium problem by virtue of setting F(x,y) = max i=1,...,p[fi(y)−fi(x)]. •The Stampacchia variational inequality problem demands finding ¯ x∈Ssuch that <G(¯ x),y−¯ x>≥0, ∀y∈S where G:Rp→Rp and S⊆Rp is a closed set. This problem is also an equilibrium problem where F(x,y) =<G(x),y−x>. • Nash equilibrium problems in a non-cooperative game with p players where each player i has a set of possible strategies Ki⊆Rni aim to minimize a loss function fi:K→R with K=K1×. . . ×Kp . Thus, a Nash equilibrium point is any ¯ x∈K such that no player can reduce its loss by unilaterally changing their strategy, i.e., any ¯ x∈Ksuch that fi(¯ x)≤fi(¯ x(yi)) holds for any yi∈Ki for any i= 1, . . . , p , with ¯ x(yi)) denoting the vector obtained from ¯ x by replacing ¯ xiwith yi. Therefore, this problem amounts to solving an equilibrium problem with F(x,y) = p ∑ i=1 [fi(x(yi)) −fi(x)]. Despite their apparent diversity, all the above-mentioned problems can be framed as particular cases of the vector equilibrium problem and thus can all be encompassed in a single mathematical picture. Due to the power of this formulation, it is of great interest to obtain and study the Karush–Kuhn–Tucker (KKT) optimality conditions for the solution of such, more general problems. Thanks to their capacity to provide such a fundamental insight, vector equilibrium problems are an active branch of non-linear analysis with plenty of publications being made up to this date. For example, in 2003, authors such as Iusem and Sosa [ 2 ] studied the relation between equilibrium problems and some auxiliary convex problems. In addition, over the past century, the field of physics departed from euclidean geometry as a space in which to allocate its theories, opting instead for more complex spaces also known as manifolds. A historical landmark that illustrates this example is Einstein’s theory of gravity that revolves around the concept of space-time curvature on a Riemannian manifold. Other less known but equally fundamental applications in the fields of physics involve the appearance of symplectic manifolds in the treatment of Hamiltonian vector fields or Noether’s theorem. Smooth Riemannian manifolds are spaces that contain curvature, as opposed to Euclidean spaces which are flat everywhere. This can be mathematically expressed as ax +by /∈M , ∀x , y∈M , a , b∈R , where M is a Riemannian manifold. Nonetheless, Riemannian geometry constitutes a generalization of the Euclidean case. This can be easily understood by introducing the notion of tangent planes. For any point of a smooth curved space, say a 2-Sphere, it is always possible to define a flat tangent plane to that point; i.e., a Euclidean space. We can think of this in the same way we think of the Earth to be flat at local scales while overall being spherical. Indeed, all curved manifolds locally resemble Euclidean space, which is a vital property for our understanding of them. However, cartography can empirically tell us that flat projections of curved surfaces onto planes fails to faithfully represent the real dimensions of the objects that live on the original curved surface especially at large scales Symmetry 2019,11, 1037 3 of 12 where the locality condition starts weakening. Thus, metricity is no longer trivial and measurements of distances need to account for such curvature. At this point, we can already see how Euclidean spaces are simply Riemannian manifolds for which the tangent plane to any of its points is identical to the plane itself. Thus, in Euclidean spaces, vectors living of the surface are equivalent to vectors living on its tangent space. It is this key feature of Euclidean geometry that allows for the simple definition of distance as the dot product. Thus, given a vector u , if allocated in an Euclidean space, its length is given by |u|2=<u , u> . On the other hand, in non-flat spaces it is necessary to account for the distortion of the distances when projected to the tangent space. Riemannian manifolds are those equipped with a so called “metric tensor”; commonly denoted kij , that allows us to adequately define distances; i.e., |u|2=kijuiuj . (see Section 2 for more details). This new definition of length has direct short comings in minimization and equilibrium. The Euclidean line element, the shortest connection between two points on a flat surface, is replaced on manifolds by a geodesic equation which plays the role of straight lines in non-flat spaces. This can be seen from the fact that geodesic curves are solutions to the Euler–Lagrange equations which minimize the functional of the Lagrangian given by the metric of such space, L=kijdxidxj , and as such describe the trajectories that minimize the action necessary to move from A to B. For example, the orbits of planets obey geodesics despite clearly not being straight in a Euclidean sense. A Hadamard manifold is a simply connected complete Riemannian manifold of non-positive sectional curvature. The motivation of the study of Hadamard spaces is that they share some properties with Euclidean spaces. One of them is the separation theorem (see Ferreira and Oliveira [3]). In addition, for any two points in M , there exists a minimal geodesic joining these two points. In a Hadamard manifold, the geodesic between any two points is unique and the exponential map at each point of M is a global diffeomorphism. Moreover, the exp map is defined on the whole tangent space ([4]). However, the minimization of functions on a Hadamard manifold is locally equivalent to the smoothly constrained optimization problem on a Euclidean space, due to the fact that every C∞ Hadamard manifold can be isometrically embedded in an Euclidean space by virtue of John Nash’s embedding theorem. This is consistent with the intuition we previously laid out. The study of optimization problems on Hadamard manifolds is a powerful tool. This is due to the fact that, generally, solving nonconvex constrained problems in Rn with the Euclidean metric can be also framed as solving the unconstrained convex minimization problem in the Hadamard manifold feasible set with the affine metric (see [ 5 ]). In Colao et al. [ 5 ] the existence of solutions for equilibrium problems under some suitable conditions on Hadamard manifolds and their applications to Nash equilibrium for non-cooperative games was studied. In the same way, in Németh [ 6 ] the existence and uniqueness results for variational inequality problems on Hadamard manifolds were obtained. Moreover, many optimization problems cannot be solved in linear spaces, for example, controlled thermonuclear fusion research (see [ 7 ]), signal processing, numerical analysis and computer vision (see [ 8 , 9 ]) require Hadamard manifold structures for their modeling. Also, geometrical structures hidden in data sets of machine learning problems are studied in terms of manifolds. In the field of medicine, Hadamard manifolds have been used in the analysis of magnetic resonances to quantify the growth of tumors and consequently deduce their state of progression, as shown by Fletcher et al. [10] . The geometry necessary to understand and perform these techniques is best understood through the use of manifolds and symmetric structures. For example, the set of symmetric positive definite matrices used in magnetic resonance imaging to study Alzheimer’s disease [ 11 ] is one case in which this translation to manifolds is necessary. In addition, other problems in computer vision, signal processing or learning algorithms employ geodesic curves when addressing optimization problems. Finally, in economics, the search of Nash–Stampacchia equilibria points using Hadamard manifolds has been used by Kristály [12]. Symmetry 2019,11, 1037 4 of 12 It is known that a convex environment has good properties for the search of optimal points. In Ferreira [ 13 ], the author gives necessary and sufficient conditions for convex functions on Hadamard manifolds. A significant generalization of the convex functions are the invex functions, introduced by Hanson [ 14 ], where the x-yvector is replaced by any function η(x , y) . The main result of invex functions states that a scalar function is invex if and only if every critical point is a global minimum solution. This property is essential to obtain optimal points through algorithms, due to the coincidence of critical points and solutions being always assured. In Barani and Pouryayeli [ 15 ] and Hosseini and Pouryayevali [ 16 ], the relation between invexity and monotonicity using the mean value theorem is studied. Ruiz-Garzón et al. [ 17 ] showed that invexity can be characterized in the context of Riemannian manifolds for both scalar and vector cases, in a similar way to Euclidean spaces. Recently, in Ahmad et al. [18] the authors introduced the log-preinvex and log-invex functions on Riemannian manifolds and the mean value theorem on Cartan-Hadamard manifolds. In the same way, several authors have studied vector equilibrium problems. Ansari and Flores-Bazán [ 19 ] were capable of providing a theorem of existence of solutions to vector quasi-equilibrium problems. Furthermore, a characterization for a weakly efficient Pareto point for the vector equilibrium problems with constraints under convexity conditions on real Hausdorff topological vector spaces were presented by Gong [ 20 ]. In the following years, scalarization results for the solutions to the vector equilibrium problems were also given by Gong [ 21 ]. Later, optimality conditions for weakly efficient Pareto points to vector equilibrium problems with constraints in real normed spaces were investigated by Wei and Gong [ 22 ]. Also, sufficient conditions of weakly efficient Pareto points on real Banach spaces for vector equilibrium and vector optimization problems with constraints under generalized invexity were obtained by Feng and Qiu [23]. Motivated by Gong’s works mentioned above, our objective will focus on extending the KKT necessary and sufficient conditions for constrained vector equilibrium problems obtained in topological or normed spaces to other environments like the Hadamard manifolds, not present in the literature up to date of publication. Hence, we propose a generalization that extends the linear space definition to Hadamard manifolds, by virtue of substituting line segments by geodesic arcs. We will see that the KKT classic conditions for constrained vector optimization are a particular case of the ones obtained for constrained vector equilibrium problem. The organization of the paper is as follows: In Section 2, we discuss notation, differentials and invex function concepts on Hadamard manifolds. Section 3is devoted to proving the main results obtained in this paper, and studying the necessary and sufficient optimality conditions for weakly efficient points of the constrained vector equilibrium problem. Section 4dwells on how the previous results can be reduced to classical KKT conditions for constrained vector optimization problems, first obtained by William Karush [ 24 ] and rediscovered by Harold Kuhn and Albert Tucker [ 25 ]. Finally, an example is presented as well as the final conclusions. 2. Preliminaries Let M be a C∞ -manifold modeled on a Hilbert space H endowed with a Riemannian metric gx on a tangent space TxM . We denote by TxM the tangent space of M at x , by TM =Sx∈MTxM the tangent bundle of M , by ¯ TM an open neighborhood of the submanifold M of TM . The corresponding norm is denoted by k.kxand the length of a piecewise C1curve α:[a,b]→Mis defined by L(α) = Zb akα0(t)kα(t)dt. We define das the distance which induces the original topology on Msuch that d(x,y) = inf{L(α)|αis a piecewise C1curve joining xand y∀x,y∈M}. Symmetry 2019,11, 1037 5 of 12 If d is the distance induced by the Riemannian metric kij then any Riemannian manifold (M , kij) can be converted into a metric space (M , d) . The derivatives of the curves at a point x on the manifold lies in a vector space TxM. Whatever path αjoining xand yin Msuch that L(α) = d(x,y)is a geodesic. Let exp : ¯ TM →M be the Riemannian exponential map defined as expx(V) = αV( 1 ) for every V∈¯ TM, where αVis the geodesic starting at xwith velocity V(i.e., α(0) = x,α0(0) = V). Assume now that ηis a map η:M×M→TM defined on the product manifold such that η(x,y)∈TyM,∀x,y∈M. Definition 1. [ 26 ] A subset S1 of M is considered totally convex if S1 contains every geodesic αx,y of M whose endpoints x and y belong to S1. On a Hadamard manifold M , we can define the function η as η(x , y) = α0 x,y( 0 ) for all x , y∈M . This function plays the same role of x−y∈Rn . Here αx,y is the unique minimal geodesic joining y to xas follows αx,y=expy(λexp−1 yx)∀λ∈[0, 1]. Example 1. Let M=R++ ={y∈R:y> 0 } endowed with the Riemannian metric defined by g(y) = y−2 be a Hadamard manifold. Hyperbolic spaces and geodesic spaces, more precisely, a Busemann non-positive curvature (NPC) space are examples of Hadarmard manifolds. We will need an adequate concept of the differential: Definition 2. [ 27 ] A mapping fi:M→R is said to be a differential map along the geodesic αx,y at y∈M if and only if the limit f0 i(y) = lim λ→0 fi(expy(λη(x,y))) −fi(y) λkη(x,y)k exists. The gradient of a real-valued C∞ function f= (f1 , . . . fp):S1⊆M→Rn on M in x , denoted by gradfx= (f0 1(x) , f0 2(x) , . . . , f0 n(x)) , is the unique vector in TxM such that d fx(X) = hgrad fx , Xi for all X in TxM is the differential of f at ¯ x of X. Remark 1. The differential of f at ¯ x of X is similar to the definition of directional derivative in the Euclidean space. Let S1⊂M be a nonempty open totally convex subset and let F:S1×S1→Rp , g:S1→Rp be mappings. Definition 3. We define the constraint set S={x∈S1:g(x)∈ −Rp +} and consider the vector equilibrium problem with constraints (VEPC): find x ∈S such that F(x,y)/∈ −Rp +\ {0},∀y∈S where Rp +is the non-negative orthant of Rp. We recall the classical concept: Definition 4. A vector x∈S satisfying F(x , y)/∈ −int Rp + , ∀y∈S is called a weakly efficient Pareto point to the VEPC. Notation 1. We denote as Hx(y) = F(x,y),∀y∈S1,given x ∈S, where H :S1→Rpis a mapping. Symmetry 2019,11, 1037 6 of 12 Inspired by the concept of convexity on a linear space, the notion of invexity function concept on Hadamard manifolds has become a successful tool in vector optimization. This generalized definition was notably provided by Hanson in [14]. Definition 5. Let S1 be a nonempty open totally convex subset of a Hadamard manifold M . A differentiable h:S1→Rp function is said to be a Rp + -invex at ¯ x∈S1 respect to η:M×M→TM if there exist η(x,¯ x)∈T¯ xM such that h(x)−h(¯ x)−dh ¯ x(η(x,¯ x)) ∈Rp +. Using the previously stated definitions, we can obtain the sufficient conditions for optimality by virtue of the assumption of invexity of the functions of the problem. 3. Main Results Next, we will obtain a characterization for the weakly efficient points of VEPC through the application of necessary and sufficient optimality conditions. We start with the necessary conditions: Theorem 1. [Necessary KKT-conditions] Let S1 be a nonempty open totally convex subset of a Hadamard manifold M and let F:S1×S1→Rp , g:S1→Rp , η:M×M→TM be mappings. Let F(¯ x , ¯ x) = H¯ x(¯ x) = 0. Assume that H and g are differentiable at ¯ x∈S . Furthermore, assume that there exists x1∈S1 such that g(¯ x) + dg¯ x(η(x1 , ¯ x)) ∈ −int Rp + . If ¯ x is a weakly efficient Pareto point to the VEPC, then there exists v ∈Rp +\ {0}, u ∈Rp +such that vdH¯ x(η(x,¯ x)) + udg ¯ x(η(x,¯ x)) ≥0, ∀x∈S1(1) ug(¯ x) = 0. (2) Proof. Let there be ¯ x∈Sas a weakly efficient Pareto point to the VEPC. We denote by W={(y,z)∈Rp×Rp: there exists x∈S1, such that y−dH¯ x(η(x,¯ x)) ∈int Rp +, z−[g(¯ x) + dg¯ x(η(x,¯ x))] ∈int Rp +}. It may be noted that W is a nonempty open totally convex set. This proof can be divided into five steps: Step 1. We have to prove that ( 0, 0 )/∈W . By reduction ad absurdum, if ( 0, 0 )∈W⇒ ∃x0∈S1 , such that dH¯ x(η(x0,¯ x)) ∈ −int Rp +,g(¯ x) + dg¯ x(η(x0,¯ x)) ∈ −int Rp +. (3) From the differentiability we obtain that dH¯ x(η(x0,¯ x)) = lim λ→0 1 λ[H¯ x(exp ¯ x(λη(x0,¯ x)) −H¯ x(¯ x)] ∈ −int Rp +(4) g(¯ x) + dg¯ x(η(x0,¯ x)) = g(¯ x) + lim λ→0 1 λ[g(exp ¯ x(λη(x0,¯ x))) −g(¯ x)] ∈ −int Rp +. (5) As −int Rp +is an open set, then ∃λ0, 0 <λ0<1 such that 1 λ0 [H¯ x(exp ¯ x(λ0η(x0,¯ x))) −H¯ x(¯ x)] ∈ −int Rp +(6) g(¯ x) + 1 λ0 [g(exp ¯ x(λ0η(x0,¯ x))) −g(¯ x)] ∈ −int Rp +. (7) Symmetry 2019,11, 1037 7 of 12 By hypothesis, from g(¯ x)∈ −Rp +,F(¯ x,¯ x) = H¯ x(¯ x) = 0, and 1 λ0 >1, then H¯ x[exp ¯ x(λ0η(x0,¯ x))] ∈ −int Rp +and g(exp ¯ x(λ0η(x0,¯ x)))∈ −int Rp +. (8) As S1is a totally convex set we have that exp ¯ x(λ0η(x0,¯ x)) ∈S1,F(¯ x,exp ¯ x(λ0η(x0,¯ x)))∈ −int Rp +(9) and g(exp ¯ x(λ0η(x0,¯ x)))∈ −int Rp +(10) stands in contradiction with ¯ x∈S as a weakly efficient Pareto point to the VEPC, consequently (0, 0)/∈W. Step 2. We will prove that there exists a multiplier v∈Rp + . As W is an open set and the separation theorem holds (see Theorem 2.13 and Remark 2.14 in [ 28 ]) or [ 3 ]), there exists (v , u)6= ( 0, 0 )∈Rp×Rp such that vy +uz >0, ∀(y,z)∈W. (11) Let (y,z)∈Wbe a point then ∃x∈S1such that y−dH¯ x(η(x,¯ x)) ∈int Rp +,z−[g(¯ x) + dg¯ x(η(x,¯ x))] ∈int Rp +. (12) For any r∈int Rp +,s∈int Rp +,t0,t00 >0, we have (y+t0r,z)∈Wand (y,z+t00s)∈W. From Equation (11) we have that v(y+t0r) + u(z)>0, ∀r∈int Rp +,t0>0. (13) Then vr >−uz −vy t0. (14) Letting t0→∞ we get vr ≥ 0, ∀r∈int Rp + and therefore vr ≥ 0 for all r∈Rp + , that is v∈Rp + . In the same way, we can show that u∈Rp +. Step 3. We will prove that v6= 0, thus is, v∈Rp +\ { 0 } . By reduction ad absurdum, if v= 0, from Equation (11) we get uz >0, ∀(y,z)∈W. (15) According to the hypothesis, ∃x1∈S1 such that g(¯ x) + dg¯ x(η(x1 , ¯ x)) ∈ −int Rp + ; then, we obtain (dH¯ x(η(x1,¯ x)) + r,g(¯ x) + dg¯ x(η(x1,¯ x)) + s)∈W,∀r∈int Rp +∀s∈int Rp +. (16) Therefore, from Equation (11) we have that u[g(¯ x) + dg¯ x(η(x1,¯ x)) + s]>0, ∀s∈int Rp +(17) us >−u[g(¯ x) + dg¯ x(η(x1,¯ x))]. (18) As [g(¯ x) + dg¯ x(η(x1 , ¯ x))] ∈ −int Rp + , and if s= 0, we get u· 0 = 0 > 0, which implies a contradiction, thus v6=0. Step 4. We will prove the first KKT condition. Since (dH¯ x(η(x,¯ x)) + r,g(¯ x) + dg¯ x(η(x,¯ x)) + s)∈W,x∈S1,r∈int Rp +,s∈int Rp +. (19) Symmetry 2019,11, 1037 8 of 12 From Equation (11) we get v[dH¯ x(η(x,¯ x)) + r] + u[g(¯ x) + dg¯ x(η(x,¯ x)) + s]>0, ∀x∈S1,r∈int Rp +,s∈int Rp +. (20) Letting r→0, s→0, we obtain vdH¯ x(η(x,¯ x)) + u[g(¯ x) + dg¯ x(η(x,¯ x))] ≥0, ∀x∈S1. (21) Step 5. We will prove the second KKT condition. As dH¯ x(η(¯ x,¯ x)) + t0r,g(¯ x) + dg¯ x(η(¯ x,¯ x)) + t0s∈W,∀r∈int Rp +,s∈int Rp +,t0>0. (22) From Equation (11) we have that v[dH¯ x(η(¯ x,¯ x)) + t0r] + u[g(¯ x) + dg¯ x(η(¯ x,¯ x)) + t0s] = t0vr +ug(¯ x) + t0us >0. (23) Letting t0→ 0, we obtain ug(¯ x)≥ 0. Noting that g(¯ x)∈ −Rp + and u∈Rp + , we have that ug(¯ x)≤0, in consequence ug(¯ x) = 0 (24) and therefore ∃v∈Rp +\ {0},u∈Rp +such that KKT conditions vdH¯ x(η(x,¯ x)) + udg ¯ x(η(x,¯ x)) ≥0, ∀x∈S1(25) ug(¯ x) = 0 (26) hold. Let us see now the reciprocal of the previous theorem. To obtain it we first need conditions of invexity. Theorem 2. [Sufficient KKT-conditions] Let S1 be a nonempty open totally convex subset of Hadamard manifold M and let F:S1×S1→Rp , g:S1→Rp be mappings. Let F(¯ x , ¯ x) = H(¯ x) = 0. Assume that H and g are differentiable at ¯ x∈S . H and g are Rp + -invex at ¯ x respect to η on S1 . If there exist v∈Rp +\ { 0 } and u∈Rp + such that vdH¯ x(η(x,¯ x)) + udg ¯ x(η(x,¯ x)) ≥0, ∀x∈S1(27) ug(¯ x) = 0 (28) then ¯ x is a weakly efficient Pareto point to the VEPC. Proof. On the assumption that Hand gare Rp +-invex at ¯ xrespect to ηon S1then dH¯ x(η(x,¯ x)) ∈H¯ x(x)−H¯ x(¯ x)−Rp +=H¯ x(x)−Rp +,∀x∈S1(29) dg¯ x(η(x,¯ x)) ∈g(x)−g(¯ x)−Rp +,∀x∈S1. (30) From v∈Rp +\ {0},u∈Rp +and (27) we obtain that vH¯ x(x) + u(g(x)−g(¯ x)) = vdH¯ x(η(x,¯ x)) + udg ¯ x(η(x,¯ x)) ≥0, ∀x∈S1. (31) From hypothesis (28), we get on the one hand that: vH¯ x(x) + ug(x)≥0, ∀x∈S1. (32) Symmetry 2019,11, 1037 9 of 12 On the other hand, we will show that ¯ x is a weakly efficient Pareto point to the VEPC. If not, consequently by definition ∃y0∈Ssuch that F(¯ x,y0)∈ −int Rp +. (33) From v∈Rp +\ {0} ⇒ vF(¯ x,y0)<0. Since y0∈S, we have g(y0)∈ −Rp +, so ug(y0)≤0 because of u∈Rp +and then vF(¯ x,y0) + ug(y0)<0 (34) stands in contradiction with (32) and therefore ¯ xis a weakly efficient Pareto point to the VEPC. Remark 2. Theorem 3.1 in [ 20 ] on real Hausdorff topological vector spaces and Theorem 3.2 and Theorem 3.4 in [ 22 ] on real normed spaces are particular cases of Theorems 1and 2obtained in this paper on Hadamard manifolds. The same is true for Theorems 3.1 and 3.3 in [23] on real Banach spaces. To sum up, we obtain the KKT optimality conditions for weakly efficient Pareto points to the vector equilibrium problems with constraints. These results are not only necessary but also sufficient. 4. Application As a particular case of the results obtained in the previous section, we will obtain the optimality conditions of KKT for constrained vector optimization problems. Let us consider the constrained multiobjective programming (CVOP) defined as: (CVOP) min f(x) subject to: g(x)≤0 x∈X⊆M where f= (f1 , . . . fp):X⊆M→Rp , g= (g1 , . . . , gm):X⊆M→Rm are differentiable multiobjective functions on the open set X⊆Mand let Mbe a Hadamard manifold. As a consequence of the previous theorems and considering CVOP as a particular case of VEPC we have the KKT classical conditions. Corollary 1. Let S1 be a nonempty open totally convex subset of Hadamard manifold M and let f , g:S1→Rp be mappings. Assume that f and g are differentiable at ¯ x∈S . Furthermore, assume that there exists x1∈S1 such that g(¯ x) + dg¯ x(η(x1 , ¯ x)) ∈ −int Rp + . If ¯ x is a weakly efficient Pareto point to the CVOP, then there exist v∈Rp +\ {0}, u ∈Rp +such that vd f ¯ x(η(x,¯ x)) + udg ¯ x(η(x,¯ x)) ≥0, ∀x∈S1(35) ug(¯ x) = 0. (36) Corollary 2. Let S1 be a nonempty open totally convex subset of Hadamard manifold M and let f , g:S1→Rp be mappings. Assume that f and g are differentiable at ¯ x∈S . Assume that f and g are differentiable at ¯ x∈S and f and g are Rp +-invex respect at ¯ x to ηon S1. If there exist v ∈Rp +\ {0}, u ∈Rp +such that vd f ¯ x(η(x,¯ x)) + udg ¯ x(η(x,¯ x)) ≥0, ∀x∈S1(37) ug(¯ x) = 0 (38) then ¯ x is a weakly efficient Pareto point to the CVOP.