Tangencies and Polynomial Optimization

Given a polynomial function $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ and a unbounded basic closed semi-algebraic set $S \subset \mathbb{R}^n,$ in this paper we show that the conditions listed below are characterized exactly in terms of the so-called {\em tangency variety} of $f$ on $S$: (i) The $f$ is bounded from below on $S;$ (ii) The … Read more

On the local stability of semidefinite relaxations

In this paper we consider a parametric family of polynomial optimization problems over algebraic sets. Although these problems are typically nonconvex, tractable convex relaxations via semidefinite programming (SDP) have been proposed. Often times in applications there is a natural value of the parameters for which the relaxation will solve the problem exactly. We study conditions … Read more

The Many Faces of Degeneracy in Conic Optimization

Slater’s condition — existence of a “strictly feasible solution” — is a common assumption in conic optimization. Without strict feasibility, first-order optimality conditions may be meaningless, the dual problem may yield little information about the primal, and small changes in the data may render the problem infeasible. Hence, failure of strict feasibility can negatively impact … Read more

Scenario Tree Reduction Methods Through Changing Node Values

To develop practical and efficient scenario tree reduction methods, we introduce a new methodology which allows for changing node values, and an easy-to-calculate distance function to measure the difference between two scenario trees. Based on minimizing the new distance, we first construct a primitive scenario tree reduction model which also minimizes the Wasserstein distance between … Read more

Coercive polynomials: Stability, order of growth, and Newton polytopes

In this article we introduce a stability concept for the coercivity of multivariate polynomials $f \in \mathbb{R}[x]$. In particular, we consider perturbations of $f$ by polynomials up to the so-called degree of stable coercivity, and we analyze this stability concept in terms of the corresponding Newton polytopes at infinity. For coercive polynomials $f \in \mathbb{R}[x]$ … Read more

On Cournot-Nash-Walras equilibria and their computation

This paper considers a model of Cournot-Nash-Walras (CNW) equilibrium where the Cournot-Nash concept is used to capture equilibrium of an oligopolistic market with non-cooperative players/ rms who share a certain amount of a so-called rare resource needed for their production, and the Walras equilibrium determines the price of that rare resource. We prove the existence of … Read more

Stability and genericity for semi-algebraic compact programs

In this paper we consider the class of polynomial optimization problems with inequality and equality constraints, in which every problem of the class is obtained by perturbations of the objective function, while the constraint functions are kept fixed. Under certain assumptions, we establish some stability properties (e.g., strong H\”older stability with explicitly determined exponents, semicontinuity, … Read more

Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods

The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, … Read more

Partial Second-Order Subdifferentials in Variational Analysis and Optimization

This paper presents a systematic study of partial second-order subdifferentials for extended-real-valued functions, which have already been applied to important issues of variational analysis and constrained optimization in finite-dimensional spaces. The main results concern developing extended calculus rules for these second-order constructions in both finite-dimensional and infinite-dimensional frameworks. We also provide new applications of partial … Read more

On the Dynamic Stability of Electricity Markets

In this work, we present new insights into the dynamic stability of electricity markets. In particular, we discuss how short forecast horizons, incomplete gaming, and physical ramping constraints can give rise to stability issues. Using basic concepts of market efficiency, Lyapunov stability, and predictive control, we construct a new stabilizing market design. A numerical case … Read more