ABSTRACT In this paper, we present an approximate proximal point method for addressing the variational inequality problem on Hadamard manifolds, and we analyse its convergence properties. The proposed algorithm exhibits inexactness in two aspects. Firstly, each proximal subproblem is approximated by utilizing the enlargement of the vector field under consideration, and subsequently, the next iteration is obtained by solving this subproblem while allowing for a suitable error tolerance. As an illustrative application, we develop an approximate proximal point method for nonlinear optimization problems on Hadamard manifolds.

The optimization problem concerning the determination of the minimizer for the sum of convex functions holds significant importance in the realm of distributed and decentralized optimization. In scenarios where full knowledge of the functions is not available, limiting information to individual minimizers and convexity parameters – either due to privacy concerns or the nature of solution analysis – necessitates an exploration of the region encompassing potential minimizers based solely on these known quantities. The characterization of this region becomes notably intricate when dealing with multivariate strongly convex functions compared to the univariate case. This paper contributes outer and inner approximations for the region harboring the minimizer of the sum of two strongly convex functions, given a constraint on the norm of the gradient at the minimizer of the sum. Notably, we explicitly delineate the boundaries and interiors of both the outer and inner approximations. Intriguingly, the boundaries as well as the interiors turn out to be identical. Furthermore, we establish that the boundary of the region containing potential minimizers aligns with that of the outer and inner approximations.

Metric fixed point theory is becoming increasingly significant across various fields, including data science and iterative methods for solving optimization problems. This paper aims to introduce new fixed point theorems for set-valued mappings under novel regularity conditions, such as orbital regularity and orbital pseudo-Lipschitzness. Instead of traditional metric spaces, we adopt the framework of quasi-metric spaces, motivated by the need to address problems in spaces that are not necessarily metric, such as function spaces of homogeneous type. We also explore the stability of the set of fixed points under variations of the set-valued mapping. Additionally, we provide estimates for the distances from a given point to the set of fixed points and between two sets of fixed points. Building on these findings, we extend the discussion to similar problems involving fixed, coincidence, and cyclic/double fixed points within this framework. Our results generalize recent findings from the literature, including those in Ait Mansour M, Bahraoui MA, El Bekkali A. [Metric regularity and Lyusternik-Graves theorem via approximate fixed points of set-valued maps in noncomplete metric spaces. Set-Valued Var Anal. 2022;30(1):233–256. doi: 10.1007/s11228-020-00553-1], Dontchev AL, Rockafellar RT. [Implicit functions and solution mappings. a view from variational analysis. Dordrecht: Springer; 2009. Springer Monographs in Mathematics], Ioffe AD. [Variational analysis of regular mappings. Springer, Cham; 2017. Springer Monographs in Mathematics; theory and applications. doi: 10.1007/978-3-319-64277-2], Lim TC. [On fixed point stability for set-valued contractive mappings with applications to generalized differential equations. J Math Anal Appl. 1985;110(2):436–441. doi: 10.1016/0022-247X(85)90306-3] and Tron NH. [Coincidence and fixed points of set-valued mappings via regularity in metric spaces. Set-Valued Var Anal. 2023;31(2):22. Paper No. 17. doi: 10.1007/s11228-023-00680-5].

ABSTRACT In this paper, we consider convex underestimators for univariate nonconvex functions over an interval in R . We propose a new convex underestimator that can be used for computing the lower bound of the range of nonconvex functions, or for solving global optimization problems. We show that the new underestimator is tighter than other underestimators which are developed in the literature.

ABSTRACT This paper focuses on the investigation of null controllability in the context of stochastic controlled relaxed systems with one control, aiming to establish several equivalent conditions for this property. More precisely, by the classical duality analysis, we present the equivalence between an observability inequality for the backward stochastic system associated with the relaxed system and null controllability. Furthermore, we characterize the null controllability through a minimal norm control problem, employing a constructed quadratic functional and leveraging a variational characterization of its minimizer. Finally, we provide an application of our findings and extend the analysis to relaxed systems with two controls.

The saddle point equilibrium control problem is of great importance in differential game theory. In this paper, we investigated the optimistic value model for the saddle point equilibrium control problem based on multi-dimensional uncertain differential equations with jumps. The equilibrium equation for the proposed model is presented. Then a linear quadratic game optimistic value model and a bang-bang game optimistic value model are discussed, respectively. Finally, some applications are analysed by the results obtained in this paper.

ABSTRACT We consider the problem of minimizing the sum of a nonsmooth function and an ℓ 0 / 1 -sparsity function, which is called the ℓ 0 / 1 -sparsity regularized nonsmooth optimization ( ℓ 0 / 1 -RNO). In this paper, by defining and analyzing two classes of stationary points, we present the first-order necessary and sufficient optimality conditions for ℓ 0 / 1 -RNO. Moreover, applying these optimality results on ℓ 0 / 1 -RNO, we derive the first-order necessary and sufficient optimality conditions for both the nonsmooth M-estimation of linear regression with positive variable sparsity, and ℓ 1 -penalty minimum of 0/1 composite optimization. Especially, we obtain that the given ℓ 1 -penalty is strongly exact penalty of 0/1 composite optimization.

The generalized Nash equilibrium problem (GNEP) is an extension of the Nash equilibrium problem (NEP), which allows the strategies and the objective function of each player to depend on the decision variables of all other players. In this paper, we consider a GNEP where the objective and constraint functions are not necessarily differentiable or convex. By proposing an alternative approach based on the equivalence with a single constrained optimization problem, we then present new sharp KKT-type necessary optimality conditions for a solution of this problem under a weak constraint qualification. We also offer an example to show that these conditions are more efficient for such problems than the previous ones. Further under weak convexity assumptions, we prove that these necessary conditions are also sufficient for optimality. Finally, we extend the definitions of some constraint qualifications from classical optimization to GNEP and study their interrelations. Throughout the paper, several examples are provided to illustrate the results.

This paper aims to solve the monotone inclusion problem, minimization problem of multiple summands and the generalized Heron problem. We present an innovative approach, the modified normal S-iteration method, designed to approximate common fixed points of nearly nonexpansive sequences and families of operators via the property ( A ) . Some deductions of our results improve some existing results in the literature. To show the applicability of our result, we give application to the inclusion problem via forward–backward splitting method version of our algorithm and minimization problem via Douglas–Rachford splitting method version of our algorithm. To demonstrate the practical utility of the algorithm, we apply it to the generalized Heron problem.

Several conjugate gradient (CG) parameters have led to promising methods for optimization problems. However, some parameters, such as the Dai-Liao (DL+) and modified Polak-Ribière-Polyak (EPRP) methods, have not yet been explored in the vector optimization setting. This paper addresses this gap by proposing the DL+ and EPRP+ methods to vector optimization. We start by developing a self-adjusting DL+ algorithm to find critical points of vector-valued functions within a closed, convex, pointed cone with a nonempty interior. The algorithm is designed to satisfy the sufficient descent condition (SDC) with a Wolfe line search; if this condition is not satisfied, the algorithm adjusts by redefining the DL+ to meet the SDC. We establish the global convergence of this algorithm under the assumption of SDC without requiring regular restarts or convexity assumption on the objective functions. Similarly, we developed the EPRP+ algorithm using the same approach as the DL+ algorithm. Furthermore, under exact line search, the DL+ and EPRP+ methods reduce to Hestenes-Stiefel (HS) and Polak-Ribière-Polyak (PRP) methods, respectively. We present numerical experiments on some selected test problems sourced from the multiobjective optimization literature that demonstrate the effectiveness and practical implementation of the proposed algorithms, highlighting their promising potential.