Solutions of optimization problems on Hadamard manifolds with Lipschitz functions

symmetry S S Article Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions Gabriel Ruiz-Garzón 1,*,†,‡ , Rafaela Osuna-Gómez 2,‡ and and Antonio Rufián-Lizana 2,‡ 1Instituto de Desarrollo Social y Sostenible (INDESS), Universidad de Cádiz, 11406 Cádiz, Spain 2Departamento de Estadística e I.O., Universidad de Sevilla, 41012 Sevilla, Spain; [email protected] (R.O.-G.); [email protected] (A.R.-L.) *Correspondence: gabriel.r[email protected] † Current address: Instituto de Desarrollo Social y Sostenible (INDESS), Universidad de Cádiz, Campus de Jerez de la Frontera, Avda. de la Universidad s/n, 11405 Jerez de la Frontera, Spain. ‡ These authors contributed equally to this work. Received: 13 April 2020; Accepted: 3 May 2020; Published: 12 May 2020   Abstract: The aims of this paper are twofold. First, it is shown, for the first time, which types of nonsmooth functions are characterized by all vector critical points as being efficient or weakly efficient solutions of vector optimization problems in constrained and unconstrained scenarios on Hadamard manifolds. This implies the need to extend different concepts, such as the Karush–Kuhn–Tucker vector critical points and generalized invexity functions, to Hadamard manifolds. The relationships between these quantities are clarified through a great number of explanatory examples. Second, we present an economic application proving that Nash’s critical and equilibrium points coincide in the case of invex payoff functions. This is done on Hadamard manifolds, a particular case of noncompact Riemannian symmetric spaces. Keywords: generalized convexity; Hadamard manifold; efficient solution; vector critical point; Nash equilibrium point 1. Introduction Firstly, our area of interest is the Hadamard manifolds. This paper is concerned with the pursuit of solutions of optimization problems defined on Hadamard manifolds through critical points, where the objective function may be nonsmooth. Optimal conditions are obtained under weaker assumptions than those already existing in the literature. The idea of convex sets in a linear space is based upon the possibility of connecting any two points of the space using line segments. In nonlinear spaces such as Hadamard manifolds, linear segments are replaced by geodesic arcs. The idea behind this is the same as the one that inspired the 19th century geometricians who created non-Euclidean geometry. The use of Hadamard manifolds has the following advantages: (a) Nonconvex constrained problems in Rn are transformed into convex ones in the Hadamard manifolds (see [1]). (b) Moreover, for example, the set X={(cos t , sin t):t∈[π/ 4, 3 π/ 4 ]} is not convex in the usual sense with X⊂R2 , but X is a geodesic convex on the Poincaré upper-plane model (H2 , gH) , as it is the image of a geodesic segment (see [2]). Secondly, in this paper, we consider the concept of invexity because of the great computational advantages it offers. The optimality conditions that invexity involves are essential in obtaining optimal points through the search for critical points with practical numerical methods. The invexity concept, Symmetry 2020,12, 804; doi:10.3390/sym12050804 www.mdpi.com/journal/symmetry Symmetry 2020,12, 804 2 of 16 introduced by Hanson [ 3 ], is an extension of differentiable convexity. A scalar function is invex if and only if every critical point is a global minimum solution. From the mind of Ben-Israel and Mond [ 4 ] the pseudoinvex functions emerged and although in the scalar case these functions coincide with the invex ones in the vector case they are different (see Ruiz-Garzón et al. ([5], Example 3.2)). Thirdly, the nonsmooth optimization formulation is found to have several clear advantages over its smooth counterpart, the main one being that it produces exact solutions to optimization problems while smoothing variants only produce approximate solutions (see Li et al. [ 6 ]). The importance of generalizing optimization methods to locally Lipschitz functions lies in their applications. For example, in controlled thermonuclear fusion research [ 7 ], engineering [ 8 ], stereo vision processing [ 9 ], and machine learning or computer vision [ 10 , 11 ]. In the field of medicine, symmetric Riemannian manifolds have been used in the analysis of medical images of tumor growth, as shown by Fletcher et al. [ 12 ]. The space of diffusion tensors required in these cases is a curved manifold named as a Riemannian symmetric space. In Bejenaru and Udriste [ 13 ], the authors extended multivariate optimal control techniques to Riemannian optimization problems in order to derive a Hamiltonian approach. Finally, for this paper, special mention should be made of studies on Nash–Stampacchia equilibria. Kristály [ 2 , 14 ] studied the existence and relationship of Nash’s critical and equilibrium points using strategy sets based on geodesic convex subsets of Hadamard manifolds and convex payoff functions, taking advantage of the geometrical features of these spaces. Equilibrium theory plays a very important role within the game theory created by von Neumann and Morgenstern [ 15 ] in 1944 and the development of the “Prisoner’s Dilemma" by Tucker and Nash in 1950 [16]. The state of the art is as follows. The initial idea for this article came from a paper written by Kristály [ 2 ] in which he relates Nash’s critical points and equilibrium points under conditions of convexity. Hosseini and Pouryayevali [ 17 ] presented a subdifferential calculus for locally Lipschitz functions to prove Lebourg’s mean value theorem in Riemannian manifolds. Later, the same authors [ 18 ] obtained necessary optimality conditions for an optimization problem on complete Riemannian manifolds, but they did not obtain characterizations. Kiliçman and Saleh [ 19 ] presented a Karush–Kuhn–Tucker sufficient optimality condition as well as a new Hermite–Hadamard-type integral inequality using differentiable sub-b-s-preinvex functions. Other authors, such as Papa Quiroz and Oliveira [ 20 ], have used the concept of subdifferentials on Hadamard manifolds to prove the global convergence of their method of solving optimization problems to the critical point of a function. Bento and Cruz [ 21 ] developed a subgradient-type method for solving non-smooth vectorial optimization problems. Their method converges to a Pareto optimal point through a vector critical point on a manifold with nonnegative sectional curvature. In 2012, Colao et al. [ 1 ] proved the existence of a Nash equilibrium point on Hadamard manifolds under the condition of convexity of the payoff functions. Chen et al. [ 22 ] discussed how to obtain efficient solutions involving generalized invex functions and Karush–Kuhn–Tucker (KKT) sufficient conditions on Riemannian manifolds. In 2014, Boumal et al. [ 23 ] authored a Matlab toolbox for optimization on manifolds (www.manopt.org). An extension of optimization methods for solving minimization problems on Hadamard manifolds when the objective function is Lipschitz was proposed by Grohs and Hosseini [24]. In 2016, Gutiérrez et al. [ 25 ] provided a characterization of pseudoinvexity through the vector critical point and found efficient solutions to multiobjective optimization problems using Lipschitz functions on linear spaces. Two years later, Ruiz-Garzón et al. [ 26 ] extended these properties on Riemannian manifolds in the smooth case. In 2019, Ruiz-Garzón et al. [ 27 ] showed the existence Symmetry 2020,12, 804 3 of 16 of KKT optimality conditions for weakly efficient Pareto solutions for vector equilibrium problems, with particular focus on the Nash equilibrium problem, but only in the differential case. Contributions. The aim of our work is to characterize the types of nonsmooth functions for which the critical points are solutions to constrained and unconstrained optimization problems on Hadamard manifolds and to extend the results obtained by Gutiérrez et al. [ 25 ] and Ruiz-Garzón et al. [ 26 ] on linear spaces. For this aim, in Section 2, we introduce a number of different generalized invexity concepts (pseudoinvexity and strong pseudoinvexity, respectively) and consider the so-called generalized Jacobian, a natural subdifferential associated with a locally Lipschitz function. We illustrate these new definitions of functions with examples on Hadamard manifolds. In Section 3, the concept of pseudoinvexity allows us to determine efficient and weakly efficient Pareto solutions of an unconstrained vector optimization problem through an adequate nonsmooth vector critical point concept. As a particular case, we show that, in the scalar case and on Hadamard manifolds, the invexity and pseudoinvexity concepts coincide. In Section 4, the vector critical point and pseudoinvexity concepts are extended from unconstrained to constrained vector optimization problems. We analyze the necessary characteristics of the objective and constraint functions of a vector optimization problem so that the KKT vector critical point is an efficient and weakly efficient solution on Hadamard manifolds in the nonsmooth case. In Section 5, we prove the equivalence between Nash critical and equilibrium points with invex payoff functions. Finally, Section 6presents the conclusions to this study. 2. Preliminaries Let M be a Riemannian manifold endowed with a Riemannian metric gx on a tangent space TxM . The corresponding norm is denoted by k . kx and the length of a piecewise C1 curve α:[a , b]→M is defined by L(α) = Zb akα0(t)kα(t)dt Let dbe the distance that induces the original topology on M, defined as d(x,y) = inf{L(α)|αis a piecewise C1curve joining xand y,∀x,y∈M} It is known that any path α joining x and y in M such that L(α) = d(x , y) is a geodesic, and is called a minimal geodesic. If M is complete, then any points in M can be joined by a minimal geodesic. The derivatives of the curves at a point x on the manifold lie in a vector space TxM . We denote by TxM the n -dimensional tangent space of M at x , and denote by TM =Sx∈MTxM the tangent bundle of M . Let ¯ TM be an open neighborhood of M such that exp : ¯ TM →M is defined as expx(v) = αv( 1, x) for every v∈¯ TM , where αv is the geodesic starting at x with velocity v (i.e., α( 0 ) = x , α0( 0 ) = v ) [ 22 ]. It is easy to see that expx(tv) = αv(t,x). Let η:M×M→TM be a map defined on the product manifold such that η(x,y)∈Ty(M),∀x,y∈M. Of all the classes of Riemannian manifolds, this work is dedicated to the Hadamard manifolds. Definition 1. Recall that a simply connected complete Riemannian manifold of nonpositive sectional curvature is called a Hadamard manifold. Let M be a Hadamard manifold. Then, expx:TxM→M is a diffeomorphism, and for any two points x , y∈M , there exists a unique minimal geodesic αx,y=expx(texp−1 xy) for all t∈[ 0, 1 ] joining xto y. We now define a generalization of the concept of convex sets and convex functions in Rn: Symmetry 2020,12, 804 4 of 16 Definition 2. [ 28 ] A subset X of M is said to be a geodesic convex if, for any two points x , y∈X , the geodesic α of M has endpoints x and y belonging to X ; that is, if α:[ 0, 1 ]→M such that α( 0 ) = x and α( 1 ) = y , then α(t)∈X for all t∈[ 0, 1 ] . Furthermore, on a Hadamard manifold, X is a geodesic convex if and only if expy(texp−1 yx)∈X. Definition 3. [ 28 ] Let M be a Hadamard manifold and X⊆M be a geodesic convex. A function θ:X→R is said to be convex if, for every x,y∈X, θ(αx,y(t)) ≤t f (x) + (1−t)f(y),∀t∈[0, 1] where α(t) = expy(texp−1 yx)for every t ∈[0, 1]. Let us now recall the following concepts in the nonsmooth case. Definition 4. A real-valued function θ defined on a Hadamard manifold M is said to satisfy a Lipschitz condition of rank k on a given subset X of M if |θ(x)−θ(y)| ≤ kd(x,y)for every x,y∈X. A function θ is said to be Lipschitz near x∈M if it satisfies the Lipschitz condition of some rank on an open neighborhood of x. A function θis said to be locally Lipschitz on M if θis Lipschitz near x for every x ∈M. Example 1. The space of symmetric n×n positive-definite matrices S(n , R) endowed with the Frobenius metric defined by <U , V>=tr(U , V) is an example of Hadamard manifold. If λ1 , . . . , λn denote the n real eigenvalues of A ∈S(n,R)then λk:S(n,R)→Ris a locally Lipschitz function. With Lipschitz functions, generalized gradients or subdifferentials replace the classical derivative. Definition 5. [ 24 ] Suppose θ:M→R is a locally Lipschitz function on a Hadamard manifold M . Given another point y∈M , consider αy,v(t) = exp−1(tw) to be a geodesic passing through y with derivative w . Then, the Clarke generalized directional derivative of θ at x∈M in the direction v∈TxM , denoted by θ0(x , v) , is defined as θ0(x,v) = lim sup y→x,t↓0 θ(αy,v(t)) −θ(y) t Definition 6. We define the subdifferential of θ at x , denoted by ∂θ(x) , as the subset of TxM with the support function given by θ0(x; .), i.e., for every v ∈TxM, θ0(x,v) = sup{<A,v>:A∈∂θ(x)} It can be proved that the generalized Jacobian is ∂θ(x) = conv{lim i→∞gradθ(xi):{xi} ⊆ X,xi→x} where Xis a dense subset of Mon which θis differentiable and conv(·)denotes the convex hull. We briefly examine some particular cases. (a) When θ is a locally Lipschitz convex function, we have θ0(x ; v) = θ0(x ; v) for all x∈M . For a convex function θ:M→R , the directional derivative of θ at the point x∈M in the direction v∈TyMis defined by θ0(x,v) = lim t→0+ θ(expx(tv)) −θ(x) t Symmetry 2020,12, 804 5 of 16 and the subdifferential of θat xis ∂θ(x) = {A∈TxM|θ0(x;v)≥<A,v>,∀v∈TxM} (b) If θ is differentiable at x∈M , we define the gradient of θ as the unique vector gradθ(x)∈TxM that satisfies dθx(v) =<grad θ(x),v>∀v∈TxM However, for the vector function f= (f1 , . . . , fp):M→Rp , the generalized Jacobian ∂f(x) is contained and, in general, is different from the Cartesian product of Clarke subdifferentials of the components of f. We denote by Rp + the nonnegative orthant of Rp , and the order in Rp is defined in the usual way: y , z∈Rp , y≤z⇔z−y∈Rp + where int Rp + denotes the interior of Rp + in Rp and −int Rp + the opposite of the interior (see [29]). The notions of generalized invexity introduced by Osuna-Gómez et al. [ 30 ] for differentiable functions, and later by Gutiérrez et al. [ 31 ] for locally Lipschitz functions using the generalized Jacobian in a finite-dimensional context, can be extended to Hadamard manifolds as follows. Definition 7. Let M be a Hadamard manifold, X be an open geodesic convex subset of M , η:M×M→TM be a not necessarily differentiable function, and f:X⊆M→Rp be a locally Lipschitz function. The function f is said to be: (a) Invex (IX) at ¯ x with respect to η on X if, ∀x∈X⊆M , there exist some η(x , ¯ x)∈T¯ xM , A∈∂f(¯ x) such that f(x)−f(¯ x)−Aη(x,¯ x)∈Rp +. (b) Pseudoinvex (PIX) at ¯ x with respect to η on X if, ∀x∈X⊆M , there exist some η(x , ¯ x)∈T¯ xM , A∈∂f(¯ x)such that f(x)−f(¯ x)∈ − int Rp +⇒Aη(x,¯ x)∈ − int Rp +. (c) Strong pseudoinvex (SGPIX) at ¯ x with respect to η on X if, ∀x∈X⊆M , there exist some η(x , ¯ x)∈T¯ xM , A∈∂f(¯ x)such that f(x)−f(¯ x)∈ −Rp +\ {0} ⇒ Aη(x,¯ x)∈ − int Rp +. The function f is said to be invex (resp. pseudoinvex, strong pseudoinvex) with respect to η on X if, for every x ∈X, f is invex (resp. pseudoinvex, strong pseudoinvex) at x with respect to ηon X. The following examples illustrate the above definitions and relations on Hadamard manifolds. Example 2. Let Ω={p= (p1 , p2)∈R2:p2> 0 } be a set and let G be a 2 × 2matrix defined by G(p) = (gij(p)) with g11(p) = g22(p) = 1 p2 2 ,g12(p) = g21(p) = 0 Endowing Ω with the Riemannian metric u , v=<G(p)u , v> , we obtain a complete Riemannian manifold H2, namely, the upper half-plane model of hyperbolic space. Let f (p) = (f1,f2)(p1,p2):Ω→R2be a function with f1(p1,p2) = p1+5p2and f2(p1,p2) =      −3p1+3if p1<1 0if 1≤p1≤2 p1−2if p1>2 Symmetry 2020,12, 804 6 of 16 The function f is invex on Ω because its components are linear functions, i.e., convex and concave functions simultaneously. Example 3. Let f(p) = (f1 , f2)(p1 , p2):Ω→R2 be a function with f(p) = (|p1+p2| , −|p1+p2|) . The function f is strong pseudoinvex with respect to any η because f(p)−f(¯ p)∈ −Rp +\ { 0 } is not satisfied for p ∈Ω. Example 4. Let Ω be the upper half-plane model of hyperbolic space with the Riemannian metric u , v= <G(p)u,v>, let G be a 2×2matrix defined by G(p) = (gij(p)) with g11(p) = g22(p) = 1 p2 2 ,g12(p) = g21(p) = 0 and A=G(p)−1∇θ(p)∈∂f(p) . Let f(p) = (f1 , f2)(p1 , p2):Ω→R2 be a function defined as f1(p1,p2) = p1and f2(p1,p2) =      −p1,if p1<0 0, if 0≤p1≤1 1−p1,if p1>1 We are going to prove that f is a pseudoinvex function but not strong pseudoinvex or invex. We have that the following: (a) If p1<0or p1>1, then f0(p,v) = h p2 20 0p2 2! 1 0 −1 0 !, v1 v2!i= p2 20 −p2 20! v1 v2!= p2 2v1 −p2 2v1! (b) If p1=0or p1=1and −1≤a≤0, then f0(p,v) = h p2 20 0p2 2! 1 0 a0!, v1 v2!i= p2 20 p2 2a0! v1 v2!= p2 2v1 p2 2av1! (c) If 0<p1<1, then f0(p,v) = h p2 20 0p2 2! 1 0 0 0 !, v1 v2!i= p2 20 0 0 ! v1 v2!= p2 2v1 0! In summary, ∂f(p) =                        p2 20 −p2 20!if p1<0or p1>1 p2 20 p2 2a0!if p1=0or p1=1with −1≤a≤0 p2 20 0 0 !if 0<p1<1. The function f is pseudoinvex with respect to every η(p , p) = 3 p−p= (v1 , v2) on Ω because f(p)− f(¯ p)∈ − int R2 + implies that f should be nondecreasing, but f2 is nonincreasing and this previous condition is not satisfied. Symmetry 2020,12, 804 7 of 16 However, f is not strong pseudoinvex on Ω with respect to any η(p , ¯ p) = (v1 , v2) because we can choose p= (0, 1),¯ p= (1, 1), and A= p2 20 p2 2a0!∈∂f(1, 1) and then f(p)−f(¯ p) = f(0, 1)−f(1, 1) = (0, 0)−(1, 0) = (−1, 0)∈ −R2 +\ {(0, 0)}, but there exists no η(p,¯ p) = (v1,v2)∈T¯ pΩsuch that Aη(p,¯ p) = (p2 2v1,p2 2av1)∈ − int R2 +with −1≤a≤0. In the same manner, f is not invex on Ω because if we choose p= ( 0, 1 ) and ¯ p= ( 1, 1 ) , there exists no η(p,¯ p) = (v1,v2)∈T¯ pΩsuch that f(p)−f(¯ p)−Aη(p,¯ p) = (0, 0)−(1, 0)−(p2 2v1,p2 2av1)∈R2 +∀a∈[−1, 0]. (1) Expression (1) implies that − 1 ≥v1 and 0 ≥av1 , but for a=− 1, there is a contradiction between them. In summary, it is well known that invexity and strong pseudoinvexity imply pseudoinvexity (see [ 31 ]), but we have found that pseudoinvexity does not imply either invexity or strong pseudoinvexity. IX ⇒PIX ⇐SGPIX We now have all the tools required to discuss critical points and solutions of vector optimization problems in the next section. 3. Relations between Solutions of Vector Optimization Problems and Vector Critical Points on Hadamard Manifolds The objective of this section is to check whether nonsmooth optimality conditions obtained in linear spaces can be extended to Hadamard manifolds. In Ruiz-Garzón et al. [ 26 ], we studied the role of invexity in the scalar case on Riemannian manifolds for the differential scenario, but not that of pseudoinvexity. In this section, we study the role of pseudoinvexity in both the scalar and vector cases on the Hadamard manifolds in unconstrained VOPs when the functions are nondifferentiable. We examine when vectorial critical points coincide with efficient and weakly efficient points. In this section, we consider the unconstrained multiobjective programming problem (VOP) defined as: (VOP) min f(x) x∈X⊆M where f= (f1 , . . . fp):X⊆M→Rp , with fi:X⊆M→R for all i: 1, . . . , p , locally Lipschitz functions on the open set X⊆M, and Massumed to be a Riemannian manifold. Let us recall two classic concepts of vectorial optimization: Definition 8. A feasible point ¯ x is said to be: (a) An efficient solution for (VOP) if there does not exist another feasible point x such that f(x)−f(¯ x)∈ −Rp +\ {0}. (b) A weakly efficient solution for (VOP) if there does not exist another feasible point x such that f(x)−f(¯ x)∈ − int Rp +. Symmetry 2020,12, 804 8 of 16 We now study some relations between solutions of (VOP) and vector critical points. We will start by defining the concept of the vector critical point: Definition 9. Let M be a Hadamard manifold, X be an open geodesic convex subset of M , and f:X⊆M→ Rp be a locally Lipschitz function. A feasible point ¯ x∈X is said to be a vector critical point (VCP) with respect to η if there exist some x∈X⊆M with η(x , ¯ x)∈T¯ xM not identically zero and λ∈Rp +\ { 0 } such that λTAη(x,¯ x) = 0for some A ∈∂f(¯ x). The importance of VCPs in obtaining weakly efficient points (efficient points) can be illustrated through a characterization of pseudoinvexity (resp. strong pseudoinvexity). Theorem 1. Let M be a Hadamard manifold, X be an open geodesic convex subset of M , and f:X⊆M→Rp be a locally Lipschitz function. Every VCP with respect to η is a weakly efficient solution of (VOP) if and only if the function f is PIX with respect to the same ηon X. Proof. Firstly, we prove that fis pseudoinvex with respect to η. (a) We consider two points x , ¯ x∈X and assume that f(x)−f(¯ x)∈ − int Rp + . Then, ¯ x is not a weakly efficient solution of (VOP). By the hypothesis, we derive that ¯ x is not a VCP with respect to η , i.e., there do not exist some x∈X⊆M with η(x , ¯ x)∈T¯ xM not identically zero and λ∈Rp +\ { 0 } such that λTAη(x , ¯ x) = 0 for some A∈∂f(¯ x) . It follows from ([ 29 ], Theorem 5.1) that Aη(x,¯ x)∈ − int Rp +and fis PIX. (b) For any points x , ¯ x∈X such that f(x)−f(¯ x)/∈ − int Rp + , we define η(x , ¯ x) = 0, and therefore f is PIX with respect to ηon X. We now prove the sufficient condition. We assume by hypothesis that f is PIX with respect to η and that ¯ xis a VCP with respect to the same η. Thus, λTAη(x,¯ x) = 0 (2) for some x∈X⊆Mwith η(x,¯ x)∈T¯ xM,λ∈Rp +\ {0}, and A∈∂f(¯ x). We need to prove that ¯ x is a weakly efficient point. By reductio ad absurdum, suppose that ¯ x is not a weakly efficient solution of (VOP). Then, there exists a point x∈X such that f(x)−f(¯ x)∈ −int Rp + . Using the fact that f is PIX at ¯ x with respect to η on X , we have Aη(x , ¯ x)∈ − int Rp + , and so λTAη(x,¯ x)<0, which contradicts (2). In the same way, we can prove the following corollary. Corollary 1. Let M be a Hadamard manifold, X be an open geodesic convex subset of M , and f:X⊆M→Rp be a locally Lipschitz function. Every VCP with respect to η is an efficient solution of (VOP) if and only if the function f is strong pseudoinvex (SGPIX) with respect to ηon X. Let us underline that Theorem 1and Corollary 1show that pseudoinvexity (resp. strong pseudoinvexity) is a minimal requirement for the property that every VCP is a weakly efficient (resp. efficient) solution of problem (VOP) on a Hadamard manifold in the nonsmooth case. In summary, we have that [VCP ⇔WEff(VOP)]⇔PIX [VCP ⇔Eff(VOP)]⇔SGPIX Theorem 1extends Theorem 2.2 of Osuna et al. [ 30 ] and Theorem 5 of Gutiérrez et al. [ 25 ] from linear spaces to Hadamard manifolds. Next, an example is given to demonstrate the applicability of the previous results. Symmetry 2020,12, 804 9 of 16 Example 5. Consider the unconstrained vector optimization problem: (VOP) min f(p) = (f1,f2)(p1,p2) subject to p ∈Ω Consider the function f of Example 4. It was proved that f is pseudoinvex with respect to η(p , p) = 3 p−p on Ω={p= (p1,p2)∈R2:p2>0}. It is easy to choose some x∈X⊆M with η(x , ¯ x)∈T¯ xM not identically zero and λ∈Rp +\ { 0 } such that λTAη(x , ¯ x) = 0for some A∈∂f(¯ x) , and therefore VCP =Ω . By applying Theorem 1, we conclude that WEff(VOP) = Ω. For scalar functions, we can go one step further. Corollary 2. Assume that θ:Ω→R is locally Lipschitz and X⊂Ω is open. Then, the following statements are equivalent: (a) θis invex (IX) with respect to ηon X. (b) Every critical point (CP) of θwith respect to ηon X is a global minimum of θon X. (c) θis PIX with respect to ηon X. Proof. (a)⇒(b) If θ is IX at ¯ x , then ∀x∈X⊆M there exist some η(x , ¯ x)∈T¯ xM , A∈∂θ(¯ x) such that θ(x)−θ(¯ x)−Aη(x,¯ x)≥0 (3) If ¯ xis a VCP, then there exists some λ≥0 such that λTAη(x,¯ x) = 0 (4) for some A∈∂θ(¯ x). From (3) and (4), this implies that θ(x)−θ(¯ x)≥0 and thus, ¯ xis a global minimum. (b)⇒(a)We will prove that, ∀x∈X⊆M, there exist some η(x,¯ x)∈T¯ xM,A∈∂θ(¯ x)such that θ(x)−θ(¯ x)−Aη(x,¯ x)≥0 •Firstly, if θ(x)−θ(¯ x)<0 (5) then there exist some η(x , ¯ x)∈T¯ xM , A∈∂θ(¯ x) such that Aη(x , ¯ x)< 0. This is because, if Aη(x,¯ x)≥0, then ¯ xwill be a VCP and ¯ xis a global minimum, i.e., θ(x)−θ(¯ x)≥0, ∀x∈X which contradicts (5). Therefore, Aη(x , ¯ x)< 0. Then, as θ0(x , ·) is positively homogeneous, it follows that θ(x)−θ(¯ x)≥ Aη(x , ¯ x) , and thus θ is IX with respect to η(x , ¯ x) = tv , where t is an arbitrary positive real number. •Secondly, if θ(x)−θ(¯ x)≥0 then θis IX with respect to η(x,¯ x) = 0. (b)⇔(c)The result is given by Theorem 1. Symmetry 2020,12, 804 16 of 16 12. Fletcher, P.T.; Venkatasubramanian, S.; Joshi, S. The geometric median on Riemannian manifolds with application to robust atlas estimation. Neuroimage 2009,45, 5143–5152. [CrossRef] 13. Bejenaru A.; Udriste, C. Multivariate Optimal Control sith Payoffs defined by submanifold integrals. Symmetry 2019,11, 893. [CrossRef] 14. Kristály, A. Nash-type equilibria on Riemannian manifolds: A variational approach. J. De Mathématiques Pures Et Appliquées 2014,101, 660–688. [CrossRef] 15. Neumann, J.; Morgenstern, O. Theory of Games and Economic Behavior, 60th-Anniversary; Princeton University Press: Princeton, NJ, USA, 2007. 16. Nash, J.F. Equilibrium points in n-person games. Proc. Natl. Acad. Sci. USA 1950 ,36, 46–49. [CrossRef] [PubMed] 17. Hosseini, S.; Pouryayevali, M.R. Generalized gradients and characterization of epi-Lipschitz sets in Riemannian manifolds. Nonlinear Anal. 2011,74, 3884–3895. [CrossRef] 18. Hosseini, S.; Pouryayevali, M.R. Nonsmooth optimization techniques on Riemannian manifolds. J. Optim. Theory Appl. 2013,158, 328–342. [CrossRef] 19. Kiliçman, A.; Saleh, W. Generalized preinvex functions and their applications. Symmetry 2018 ,10, 493. [CrossRef] 20. Papa Quiroz, E.A.; Oliveira, P.R. Proximal point method for minimizing quasiconvex locally Lipschitz functions on Hadamard manifolds. Nonlinear Anal. 2012,75, 5924–5932. [CrossRef] 21. Bento, G.C.; Cruz Neto, J.X. A subgradient method for multiobjective optimization on Riemannian manifolds. J. Optim. Theory Appl. 2013,159, 125–137. [CrossRef] 22. Chen, S.I.; Huang, N.J.; O’Regan, D. Geodesic B-preinvex Functions and Multiobjective Optimization Problems on Riemannian Manifolds. J. Appl. Math. 2014. [CrossRef] 23. Boumal, N.; Mishra, B.; Absil, P.A.; Sepulchre, R. Manopt, a Matlab Toolbox for Optimization on Manifolds. J. Mach. Learn. Res. 2014,15, 1455–1459. 24. Grohs, P.; Hosseini, S. e -subgradient algorithms for locally lipschitz functions on Riemannian manifolds. Adv. Comput. Math. 2016,42, 333–360. [CrossRef] 25. Gutiérrez, C.; Jiménez, B.; Novo, V.; Ruiz-Garzón, G. Vector critical points and efficiency in vector optimization with Lipschitz functions. Optim. Lett. 2016,10, 47–62. [CrossRef] 26. Ruiz-Garzón, G.; Osuna-Gómez, R.; Rufián-Lizana, A.; Hernández-Jiménez, B. Optimality and Duality on Riemannian Manifolds. Taiwan J. Math. 2018,22, 1245–1259. [CrossRef] 27. Ruiz-Garzón, G.; Osuna-Gómez, R.; Ruiz-Zapatero, J. Necessary and Sufficient Optimality Conditions for Vector Equilibrium Problems on Hadamard Manifolds. Symmetry 2019,11, 1037. [CrossRef] 28. Udriste, C. Convex Functions and Optimization Methods on Riemannian Manifolds. In Mathematics and Its Applications; Kluwer Academic: Dordrecht, The Netherlands, 1994; Volume 297. 29. Guerraggio, A.; Luc, D.T. Optimality conditions for C 1,1 vector optimization problems. J. Optim. Theory Appl. 2001,109, 615–629. [CrossRef] 30. Osuna-Gómez, R.; Rufián-Lizana, A.; Ruiz, P. Invex Functions and Generalized Convexity in Multiobjective Programming. J. Optim. Theory Appl. 1998,98, 651–661. [CrossRef] 31. Gutiérrez, C.; Jiménez, B.; Novo, V.; Ruiz-Garzón, G. Vector variational-like inequalities and multiobjective optimization with Lipschitz functions. Appl. Math. Comput. 2015,259, 438–449. [CrossRef] 32. Osuna-Gómez, R.; Beato-Moreno, A.; Rufián-Lizana, A. Generalized Convexity in Multiobjective Programming. J. Math. Anal. Appl. 1999,233, 205–220. [CrossRef] 33. Bartl, D. Farkas’ Lemma, others theorems of the alternative, and linear programming in infinite-dimensional spaces: A purely linear-algebraic approach. Linear Multilinear Algebra 2007,55, 327–353. [CrossRef] c 2020 by the authors. Licensee MDPI, Basel, Switzerland. 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