This paper investigates the class of optimal value/marginal functions in the general framework of parametric optimization in locally convex topological vector spaces and their specializations. We address some qualitative and generalized differential properties of such functions under near convexity of their initial data. In this way, several fundamental results on nearly convex sets and functions in infinite-dimensional spaces are established and then applied to the study of optimal value functions. The obtained qualitative properties for this class include lower semicontinuity, Lipschitzian stability, near convexity, etc. Subsequently, we derive calculus rules and representation formulas for ε -subdifferentials of optimal value functions and their Fenchel conjugates in both convex and nonconvex settings. Applications of the main results are given to Fenchel and Lagrangian duality in infinite-dimensional constrained optimization by using Fenchel conjugates of set-valued mappings and the developed tools of near convex analysis.
This paper focuses on maximal marginal functions, also known as optimal value functions, which frequently arise in variational analysis, parametric optimization, and various other fields. These functions represent the value of the objective function at the optimal solution of a maximization problem, with respect to the problem’s parameters. Generally, they are inherently nonsmooth, necessitating the use of generalized differentiation techniques for their analysis. The main results of this paper provide sufficient conditions for the tangential convexity of the marginal function and offer an upper estimate of its tangential subdifferential when this property holds. The structure of the upper estimate closely resembles results already established in the literature. Examples are included to illustrate our findings.
The Variation Evolving Method (VEM), established through the variational dynamic analysis, facilitates the computation of Optimal Control Problems (OCPs) by transforming them to Initial-value Problems (IVPs) with respect to the variational time τ . In this paper, it is reformulated in the primal variable space for better performance. The Evolution Partial Differential Equation (EPDE), which theoretically guarantees the variational motion of control u towards the optimal solution, is derived in the first-order and the second-order forms. The costate-free optimality conditions are established, and the explicit analytic expressions for the costates and the Lagrange multipliers adjoining the terminal constraints are presented, in terms of the state and the control variables. Using the semi-discrete method, the EPDE is solved as a finite-dimensional IVP on Ordinary Differential Equations (ODEs), and it is proved that with a reasonable step-size, the ODE numerical solution for the transformed nonlinear stable dynamical system is precise and has an exponential convergence rate to the optimal solution.
A system of equilibrium problems is a problem consisting in finding a common solution to several equilibrium problems. In this paper, we develop algorithms based on the proximal point algorithm and the bundle concept to solve such problems. To deal with, we firstly reformulate the system of equilibrium problems as a single unconstrained equilibrium problems or with simpler constraints. Afterwards, we show that the latter is equivalent to an unconstrained optimization problem or with simpler constraints, for which we propose a general framework to construct various proximal bundle methods, where sequences of simpler optimization problems, typically quadratic programs, are to be solved. We show that our algorithms converge for every starting point, feasible or infeasible, to a solution of the system of equilibrium problems under different conditions, and for problems involving cyclically antimonotone bifunctions.
For solution mappings of parameterized models (such as optimization problems, variational inequalities, and generalized equations), standard stability (such as Aubin property) fails to hold as the parameter approaches the boundary of the feasible domain. A remedy is relative stability restricted to a constraint set (e.g., the feasible domain), which is our focus in this paper. We establish generalized differentiation criteria that characterize stability and strong stability of a solution mapping relative to a broad class of nonconvex constraint sets. Beyond this class, we give a counterexample that invalidates all known generalized differentiation criteria. Applied to generalized equations, our results yield characterizations of relative stability and relative strong stability of their solution mappings, which are further explicitly specified for affine variational inequalities. Finally, we prove a global relative stability criterion, which provides a different perspective on stability analysis and also generalizes the mean value theorem to set-valued, non-smooth mappings.
