Calmness of a perturbed Cournot Oligopoly Game with nonsmooth cost functions

This article deals with the calmness of a solution map of a Cournot Oligopoly Game with nonsmooth cost functions. The fact that the cost functions are not supposed to be differentiable allows for considering cases where some firms have diferent units of production, which have diferent marginal costs. In order to obtain results about the … Read more

Variational Analysis of the Crouzeix Ratio

Let $W(A)$ denote the field of values (numerical range) of a matrix $A$. For any polynomial $p$ and matrix $A$, define the Crouzeix ratio to have numerator $\max\left\{|p(\zeta)|:\zeta\in W(A)\right\}$ and denominator $\|p(A)\|_2$. M.~Crouzeix’s 2004 conjecture postulates that the globally minimal value of the Crouzeix ratio is $1/2$, over all polynomials $p$ of any degree and … Read more

Variational analysis of spectral functions simplified

Spectral functions of symmetric matrices — those depending on matrices only through their eigenvalues — appear often in optimization. A cornerstone variational analytic tool for studying such functions is a formula relating their subdifferentials to the subdifferentials of their diagonal restrictions. This paper presents a new, short, and revealing derivation of this result. We then … Read more

Lipschitz behavior of the robust regularization

To minimize or upper-bound the value of a function “robustly”, we might instead minimize or upper-bound the “epsilon-robust regularization”, defined as the map from a point to the maximum value of the function within an epsilon-radius. This regularization may be easy to compute: convex quadratics lead to semidefinite-representable regularizations, for example, and the spectral radius … Read more

Necessary Conditions for the Impulsive Optimal Control of Multibody Mechanical Systems

In this work, necessary conditions for the impulsive optimal control of multibody mechanical systems are stated. The conditions are obtained by the application subdifferential calculus techniques to extended-valued lower semi-continuous generalized Bolza functional that is evaluated on multiple intervals. Contrary to the approach in literature so far, the instant of possibly impulsive transition is considered … Read more

Using Simplex Gradients of Nonsmooth Functions in Direct Search Methods

It has been shown recently that the efficiency of direct search methods that use opportunistic polling in positive spanning directions can be improved significantly by reordering the poll directions according to descent indicators built from simplex gradients. The purpose of this paper is twofold. First, we analyze the properties of simplex gradients of nonsmooth functions … Read more

Stationarity and Regularity of Real-Valued Functions

Different stationarity and regularity concepts for extended real-valued functions on metric spaces are considered in the paper. The properties are characterized in terms of certain local constants. A classification scheme for stationarity/regularity constants and corresponding concepts is proposed. The relations between different constants are established. Citation University of Ballarat, School of Information Technology and Mathematical … Read more

About Regularity of Collections of Sets

The paper continues investigations of stationarity and regularity properties of set systems in normed spaces started in the previous paper of the author. It contains a summary of different characterizations (both primal and dual) of regularity and a list of sufficient conditions for a set system to be regular. Citation University of Ballarat, School of … Read more

Analysis of a Belgian Chocolate Stabilization Problem

We give a detailed numerical and theoretical analysis of a stabilization problem posed by V. Blondel in 1994. Our approach illustrates the effectiveness of a new gradient sampling algorithm for finding local optimizers of nonsmooth, nonconvex optimization problems arising in control, as well as the power of nonsmooth analysis for understanding variational problems involving polynomial … Read more

Variational Analysis of Functions of the Roots of Polynomials

The Gauss-Lucas Theorem on the roots of polynomials nicely simplifies calculating the subderivative and regular subdifferential of the abscissa mapping on polynomials (the maximum of the real parts of the roots). This paper extends this approach to more general functions of the roots. By combining the Gauss-Lucas methodology with an analysis of the splitting behavior … Read more