Estimates of Generalized Hessians for Optimal Value Functions in Mathematical Programming

We consider the optimal value function of a parametric optimization problem. A large number of publications have been dedicated to the study of continuity and differentiability properties of the function. However, the differentiability aspect of works in the current literature has mostly been limited to first order analysis, with focus on estimates of its directional derivatives and subdifferentials, given that the function is typically nonsmooth. With the progress made in the last two to three decades in major subfields of optimization such as robust, minmax, semi-infinite and bilevel optimization, and their connection to the optimal value function, there is a need for a second order analysis of the generalized differentiability properties of this function. This could enable the development of robust solution algorithms, such as the Newton method. The main goal of this paper is to provide estimates of the generalized Hessian for the optimal value function. Our results are based on two handy tools from parametric optimization, namely the optimal solution and Lagrange multiplier mappings, for which completely detailed estimates of their generalized derivatives are either well-known or can easily be obtained.

Efficient solution and value function for non-convex variational problems

In this paper, we want to investigate a wide range of non-convex variational problems and obtain some sufficient and necessary conditions for existence of a feasible solution for these problems. Hence, we define optimal value function corresponding to these problems and obtain a relationship between subdifferential of the optimal value function and the set of Lagrange multipliers.

On a Recurrence Relation Related to The Reve’s Puzzle

The 4-peg Tower of Hanoi problem, commonly known as the Reve’s puzzle, is well-known. Motivated by the optimality equation satisfied by the optimal value function M(n) satisfied in case of the Reve’s puzzle, (Matsuura et al. 2008) posed the following generalized recurrence relation T(n, a) = min {aT(n-t, a)+S(t,3)} 1≤ t ≤ n where n ≥ 1 and a ≥ 2 are integers, and S(t, 3) = 2t – 1 is the solution of the 3-peg Tower of Hanoi problem with t discs. Some local-value relationships are satisfied by T(n, a) (Majumdar et al. 2016). This paper studies the properties of T(n+1, a) – T(n, a) more closely for the case when a is an integer not of the form 2i for any integer i ≥ 2. Journal of Bangladesh Academy of Sciences, Vol. 42, No. 2, 191-199, 2018

Hadamard Directional Differentiability of the Optimal Value Function of a Quadratic Programming Problem

In this paper, we consider the stability analysis of a convex quadratic programming (QP) problem and its restricted Wolfe dual when all parameters in the problem are perturbed. Based on the continuity of the feasible set mapping, we establish the upper semi-continuity of the optimal solution mappings of the convex QP problem and the restricted Wolfe dual problem. Furthermore, by characterizing the optimal value function as a min–max optimization problem over two compact convex sets, we demonstrate the Lipschitz continuity and the Hadamard directional differentiability of the optimal value function.

An Optimal Stopping Problem for Jump Diffusion Logistic Population Model

This paper examines an optimal stopping problem for the stochastic (Wiener-Poisson) jump diffusion logistic population model. We present an explicit solution to an optimal stopping problem of the stochastic (Wiener-Poisson) jump diffusion logistic population model by applying the smooth pasting technique (Dayanik and Karatzas, 2003; Dixit, 1993). We formulate this as an optimal stopping problem of maximizing the expected reward. We express the critical state of the optimal stopping region and the optimal value function explicitly.