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This paper studies the solution stability of convex optimization and discrete convex optimal control problems in Banach spaces, where the solution set may be empty. For both the optimization problem and the optimal control problem, formulas for the ε -subdifferential of the optimal value function are derived without qualification conditions. We first calculate the ε -subdifferential of the optimal value function to a parametric optimization problem with geometrical and functional constraints. We then use the obtained results to derive a formula for computing the ε -subdifferential of the optimal value function to a discrete convex optimal control problem with linear state equations, control constraints and initial, terminal conditions. [ABSTRACT FROM AUTHOR]

Understanding the role that subgradients play in various second-order variational analysis constructions can help us uncover new properties of important classes of functions in variational analysis. Focusing mainly on the behavior of the second subderivative and subgradient proto-derivative of polyhedral functions, i.e., functions with polyhedral convex epigraphs, we demonstrate that choosing the underlying subgradient, utilized in the definitions of these concepts, from the relative interior of the subdifferential of polyhedral functions ensures stronger second-order variational properties such as strict twice epi-differentiability and strict subgradient proto-differentiability. This allows us to characterize continuous differentiability of the proximal mapping and twice continuous differentiability of the Moreau envelope of polyhedral functions. We close the paper with proving the equivalence of metric regularity and strong metric regularity of a class of generalized equations at their nondegenerate solutions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the resolvent via Busemann functions introduced by Bento, Cruz Neto, Melo (J Optim Theory Appl 195:1087–1105, 2022), and we present a proximal point method for equilibrium problems on Hadamard manifold. The resolvent in consideration is a natural extension of its counterpart in linear settings, proposed and analyzed by Combettes and Hirstoaga (J Nonlinear Convex Anal 6:117–136, 2005). The advantage of using this resolvent is that the term performing regularization is a convex function in general Hadamard manifolds, allowing us to explore the asymptotic behavior of the proximal point method to solve equilibrium problems. [ABSTRACT FROM AUTHOR]

In this paper, we first characterize generalized convex functions introduced by Linh and Penot Optimization (62: 943–959, 2013) by using generalized monotonicity of the generalized subdifferentials. We use vector variational inequalities in terms of generalized subdifferentials to identify efficient solutions of a multiobjective optimization problem involving quasiconvex functions. We also establish the Minty variational principle by utilizing the mean value theorem established by Kabgani and Soleimani-damaneh (Numer. Funct. Anal. Optim 38: 1548–1563, 2017) for quasiconvex functions in terms of Greenberg–Pierskalla subdifferentials. [ABSTRACT FROM AUTHOR]