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Very recently, P. Saha et al. [A global optimality result in probabilistic spaces using control function. Optimization. 2021;70:2387–2400] studied the existence of best proximity points for Banach type ϕ-proximal contractions in the framework of Menger probabilistic metric spaces. In this paper, we present a simple proof of their main existence result which is based on a fixed point theorem for ϕ-contractive self-mappings in complete Menger PM-spaces due to B.S. Choudhury and K. Das [A new contraction principle in Menger spaces. Acta Math Sin. 2008;24:13791386]. [ABSTRACT FROM AUTHOR]

As a complement to two recent papers by Toan and Yao (Mordukhovich subgradients of the value function to a parametric discrete optimal control problem. J Global Optim. 2014;58:595–612), and by An and Toan (Differential stability of convex discrete optimal control problems. Acta Math Vietnam. 2018;43:201–217) on subdifferentials of the optimal value function of discrete optimal control problems, this paper studies the differential stability of convex discrete optimal control problems under control constraints, where the solution set may be empty. By using a suitable sum rule for ϵ-subdifferentials and a suitable product rule for ϵ-normal directions, we obtain formulas for computing the ϵ-subdifferential of the optimal value function. Several illustrative examples are also given. [ABSTRACT FROM AUTHOR]

Linear fractional vector optimization problems are special non-convex vector optimization problems. They were introduced and first studied by E.U. Choo and D.R. Atkins in the period 1982–1984. This paper investigates the properness in the sense of Geoffrion of the efficient solutions of linear fractional vector optimization problems with unbounded constraint sets. Sufficient conditions for an efficient solution to be Geoffrion's properly efficient solution are obtained via Benson's characterization [An improved definition of proper efficiency for vector maximization with respect to cones. J Math Anal Appl. 1979;71:232–241] of Geoffrion's proper efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we consider a special implicit set-valued map representing solutions to a parametric quasi-optimization problem, (Q O p t) for short. This model finds its motivation in quasi-convex programming and generalized Nash equilibria modelled by the supremum of the so-called Nikaido–Isoda functions. We exploit a new recent variant of the celebrated Lim's Lemma considered in the context of metric regularity and approximate fixed points to establish quantitative stability for ε-approximate solutions to (Q O p t) under parametric perturbations in the spirit of the result presented for convex programming in the seminal contribution by Attouch and Wets [Quantitative stability of variational systems: III. ε-approximatesolutions. Math Program. 1993;61:197–214, Theorem 4.3]. Sharp estimates are then extended to parametric exact solutions to (Q O p t) by means of a qualitative stability analysis stressing the role of Painlevé-Kuratowski and Pompeiu-Hausdorff convergence for sets of approximate minima to a set of exact ones under usual compactness and/or completeness conditions. Finally, we apply our main result to a non-smooth mathematical program under polyhedral convex mappings and situate our contribution in the close recent literature. [ABSTRACT FROM AUTHOR]

We here adapt an extended version of the adaptive cubic regularization method with dynamic inexact Hessian information for nonconvex optimization in Bellavia et al. [Adaptive cubic regularization methods with dynamic inexact hessian information and applications to finite-sum minimization. IMA Journal of Numerical Analysis. 2021;41(1):764–799] to the stochastic optimization setting. While exact function evaluations are still considered, this novel variant inherits the innovative use of adaptive accuracy requirements for Hessian approximations introduced in the just quoted paper and additionally employs inexact computations of the gradient. Without restrictions on the variance of the errors, we assume that these approximations are available within a sufficiently large, but fixed, probability and we extend, in the spirit of Cartis and Scheinberg [Global convergence rate analysis of unconstrained optimization methods based on probabilistic models. Math Program Ser A. 2018;159(2):337–375], the deterministic analysis of the framework to its stochastic counterpart, showing that the expected number of iterations to reach a first-order stationary point matches the well-known worst-case optimal complexity. This is, in fact, still given by O (ϵ − 3 / 2) , with respect to the first-order ϵ tolerance. Finally, numerical tests on nonconvex finite-sum minimization confirm that using inexact first- and second-order derivatives can be beneficial in terms of the computational savings. [ABSTRACT FROM AUTHOR]

In this paper, we give a unified treatment of two different definitions of complementarity partition of multifold conic programs introduced independently in Bonnans and Ramírez [Perturbation analysis of second-order cone programming problems, Math Program. 2005;104(2–30):205–227] for conic optimization problems, and in Peña and Roshchina [A complementarity partition theorem for multifold conic systems, Math Program. 2013;142(1–2):579–589] for homogeneous feasibility problems. We show that both can be treated within the same unified geometric framework and extend the latter notion to optimization problems. We also show that the two partitions do not coincide, and their intersection gives a seven-set index partition. Finally, we demonstrate that the partitions are preserved under the application of nonsingular linear transformations, and in particular, that a standard conversion of a second-order cone program into a semidefinite programming problem preserves the partitions. [ABSTRACT FROM AUTHOR]

The proximal method of multipliers was proposed by Rockafellar [Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math Oper Res. 1976;1:97–116] for solving convex programming and it is a kind of proximal point method applied to convex programming. In this paper, we apply this method for solving nonlinear semidefinite programming problems, in which subproblems have better properties than those from the augmented Lagrange method. We prove that, under the linear independence constraint qualification and the strong second-order sufficiency optimality condition, the rate of convergence of the proximal method of multipliers, for a nonlinear semidefinite programming problem, is linear and the ratio constant is proportional to 1/c, where c is the penalty parameter that exceeds a threshold c ∗ > 0. Moreover, the rate of convergence of the proximal method of multipliers is superlinear when the parameter c increases to + ∞. [ABSTRACT FROM AUTHOR]

In this paper, we obtain Hölder continuity of the nonlinear scalarizing function for l-type less order relation, which is introduced by Hernández and Rodríguez-Marín (J. Math. Anal. Appl. 2007;325:1–18). Moreover, we introduce the nonlinear scalarizing function for u-type less order relation and establish continuity, convexity and Hölder continuity of the nonlinear scalarizing function for u-type less order relation. As applications, we firstly obtain Lipschitz continuity of solution mapping to the parametric equilibrium problems and then establish Lipschitz continuity of strongly approximate solution mappings for l-type less order relation, u-type less order relation and set less order relation to the parametric set optimization problems by using convexity and Hölder continuity of the nonlinear scalarizing functions. [ABSTRACT FROM AUTHOR]

In this paper, a new nonlinear scalarization function, which is a generalization of the oriented distance function, is introduced. Some properties of the function are discussed. Then the function is applied to obtain some new optimality conditions and scalar representations for set-valued vector optimization problems with set optimization criteria. In terms of the function and the image space analysis, some new alternative results for generalized parametric systems are derived. [ABSTRACT FROM AUTHOR]

A key procedure in proximal bundle methods for convex minimization problems is the definition of stability centres, which are points generated by the iterative process that successfully decrease the objective function. In this paper we study a different stability-centre classification rule for proximal bundle methods. We show that the proposed bundle variant has at least two particularly interesting features: (i) the sequence of stability centres generated by the method converges strongly to the solution that lies closest to the initial point; (ii) if the sequence of stability centres is finite,being its last element, then the sequence of non-stability centres (null steps) converges strongly to. Property (i) is useful in some practical applications in which a minimal norm solution is required. We show the interest of this property on several instances of a full sized unit-commitment problem. [ABSTRACT FROM AUTHOR]

We study semi-convex frontier (SCF) optimization problems where objective functions can be semi-convex and constraint sets can be non-polyhedron, which stem from a growing range of optimization applications such as frontier analysis, multi-objective programming in economics. The new findings of this paper can be summarized as follows: (1) We characterize non-dominated points of a non-polyhedron optimal solution set of a semi-convex frontier program. (2) We obtain optimality conditions of a constant modulus SCF program, of which the objective function is semi-convex with a constant semiconvexity modulus. (3) We obtain a non-smooth Hölder stability of the optimal solutions of a semiconvex frontier program. (4) We use generalized differentiability to establish sensitivity analysis of the optimal value function of a semi-convex frontier program. [ABSTRACT FROM AUTHOR]

This paper addresses the reverse 1-center problem on a weighted tree. Here, a facility has already been located in a predetermined node of the tree network and we want to improve the 1-center objective value at that node as efficiently as possible within a given budget. For solving this problem under uniform linear cost functions, we develop a combinatorial algorithm with running time, whereis the number of vertices of the tree. [ABSTRACT FROM AUTHOR]

In this paper, we propose a three-term conjugate gradient method via the symmetric rank-one update. The basic idea is to exploit the good properties of the SR1 update in providing quality Hessian approximations to construct a conjugate gradient line search direction without the storage of matrices and possess the sufficient descent property. Numerical experiments on a set of standard unconstrained optimization problems showed that the proposed method is superior to many well-known conjugate gradient methods in terms of efficiency and robustness. [ABSTRACT FROM AUTHOR]

In this paper, we deal with constraint qualifications, stationary concepts and optimality conditions for a nonsmooth mathematical program with equilibrium constraints (MPEC). The main tool in our study is the notion of convexificator. Using this notion, standard and MPEC Abadie and several other constraint qualifications are proposed and a comparison between them is presented. We also define nonsmooth stationary conditions based on the convexificators. In particular, we show that GS-stationary is the first-order optimality condition under generalized standard Abadie constraint qualification. Finally, sufficient conditions for global or local optimality are derived under some MPEC generalized convexity assumptions. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new simultaneous iterative algorithm for solving the split common fixed point problem of directed operators. Inspired by the idea of cyclic iterative algorithm, we also introduce two iterative algorithms which combine the process of cyclic and simultaneous together. Under mild assumptions, we prove convergence of the proposed iterative sequences. As applications, we obtain several iteration schemes to solve the inverse problem of multiple-sets split feasibility problem. Numerical experiments are presented to confirm the efficiency of the proposed iterative algorithms. [ABSTRACT FROM AUTHOR]

In this paper, new results for Hölder continuity of the unique solution to a parametric generalized vector quasiequilibrium problem are established via nonlinear scalarization, with and without using the free-disposal condition. Especially, a new kind of monotonicity hypothesis is proposed. The globally Lipschitz property together with other useful properties of the well-known Gerstewitz nonlinear scalarization function are fully exploited for proving. Moreover, our approach does not impose any convexity condition on the considered model. The oriented distance function is also employed for studying Hölder continuity. [ABSTRACT FROM AUTHOR]