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In this paper, by revisiting intersection rules for normal cones, we give formulas for estimating or computing the Fréchet/Mordukhovich/Moreau–Rockafellar subdifferentials of optimal value functions of constrained parametric optimization problems under metric qualification conditions. The results are then applied to derive chain rules for composite functions in both convex and nonconvex situations. Illustrative examples and comparisons to existing results, including those of Mordukhovich and Shao (Trans Amer Math Soc 348:1235–1280, 1996), Mordukhovich et al. (Math Program Ser B 116:369–396, 2009) and of An and Jourani (J Optim Theory Appl 192:82–96, 2022), are also addressed. [ABSTRACT FROM AUTHOR]

In this paper, a joint study of the behavior of solutions of the Heavy Ball ODE and Heavy Ball type algorithms is given. Since the pioneering work of Polyak (USSR Comput Math Math Phys 4(5):1–17, 1964), it is well known that such a scheme is very efficient for C 2 strongly convex functions with Lipschitz gradient. But much less is known when only growth conditions are considered. Depending on the geometry of the function to minimize, convergence rates for convex functions, with some additional regularity such as quasi-strong convexity, or strong convexity, were recently obtained in Aujol et al. (Convergence rates of the Heavy-Ball method for quasi-strongly convex optimization, 2020). Convergence results with much weaker assumptions are given in the present paper: namely, linear convergence rates when assuming a growth condition (which amounts to a Łojasiewicz property in the convex case). This analysis is firstly performed in continuous time for the ODE, and then transposed for discrete optimization schemes. In particular, a variant of the Heavy Ball algorithm is proposed, which converges geometrically whatever the parameters choice, and which has the best state of the art convergence rate for first order methods to minimize composite non smooth convex functions satisfying a Łojasiewicz property. [ABSTRACT FROM AUTHOR]

In the present paper, we analyze a class of convex semi-infinite programming problems with arbitrary index sets defined by a finite number of nonlinear inequalities. The analysis is carried out by employing the constructive approach, which, in turn, relies on the notions of immobile indices and their immobility orders. Our previous work showcasing this approach includes a number of papers dealing with simpler cases of semi-infinite problems than the ones under consideration here. Key findings of the paper include the formulation and the proof of implicit and explicit optimality conditions under assumptions, which are less restrictive than the constraint qualifications traditionally used. In this perspective, the optimality conditions in question are also compared to those provided in the relevant literature. Finally, the way to formulate the obtained optimality conditions is demonstrated by applying the results of the paper to some special cases of the convex semi-infinite problems. [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]

In this paper we consider a class of structured nonsmooth difference-of-convex (DC) minimization in which the first convex component is the sum of a smooth and nonsmooth functions while the second convex component is the supremum of possibly infinitely many convex smooth functions. We first propose an inexact enhanced DC algorithm for solving this problem in which the second convex component is the supremum of finitely many convex smooth functions, and show that every accumulation point of the generated sequence is an (α , η) -D-stationary point of the problem, which is generally stronger than an ordinary D-stationary point. In addition, inspired by the recent work (Pang et al. in Math Oper Res 42(1):95–118, 2017; Wen et al. in Comput Optim Appl 69(2):297–324, 2018), we propose two proximal DC algorithms with extrapolation for solving this problem. We show that every accumulation point of the solution sequence generated by them is an (α , η) -D-stationary point of the problem, and establish the convergence of the entire sequence under some suitable assumption. We also introduce a concept of approximate (α , η) -D-stationary point and derive iteration complexity of the proposed algorithms for finding an approximate (α , η) -D-stationary point. In contrast with the DC algorithm (Pang et al. 2017), our proximal DC algorithms have much simpler subproblems and also incorporate the extrapolation for possible acceleration. Moreover, one of our proximal DC algorithms is potentially applicable to the DC problem in which the second convex component is the supremum of infinitely many convex smooth functions. In addition, our algorithms have stronger convergence results than the proximal DC algorithm in Wen et al. (2018). [ABSTRACT FROM AUTHOR]