Integer packing sets form a well-quasi-ordering

An integer packing set is a set of non-negative integer vectors with the property that, if a vector x is in the set, then every non-negative integer vector y with y ≤ x is in the set as well. Integer packing sets appear naturally in Integer Optimization. In fact, the set of integer points in … Read more

On the algebraic structure of the copositive cone

We decompose the copositive cone $\copos{n}$ into a disjoint union of a finite number of open subsets $S_{\cal E}$ of algebraic sets $Z_{\cal E}$. Each set $S_{\cal E}$ consists of interiors of faces of $\copos{n}$. On each irreducible component of $Z_{\cal E}$ these faces generically have the same dimension. Each algebraic set $Z_{\cal E}$ is … Read more

Compact Representations of Structured BFGS Matrices

For general large-scale optimization problems compact representations exist in which recursive quasi-Newton update formulas are represented as compact matrix factorizations. For problems in which the objective function contains additional structure, so-called structured quasi-Newton methods exploit available second-derivative information and approximate unavailable second derivatives. This article develops the compact representations of two structured Broyden-Fletcher-Goldfarb-Shanno update formulas. … Read more

Sequential Convexification of a Bilinear Set

We present a sequential convexification procedure to derive, in the limit, a set arbitrary close to the convex hull of $\epsilon$-feasible solutions to a general nonconvex continuous bilinear set. Recognizing that bilinear terms can be represented with a finite number nonlinear nonconvex constraints in the lifted matrix space, our procedure performs a sequential convexification with … Read more

A Finitely Convergent Disjunctive Cutting Plane Algorithm for Bilinear Programming

\(\) In this paper we present and analyze a finitely-convergent disjunctive cutting plane algorithm to obtain an \(\epsilon\)-optimal solution or detect infeasibility of a general nonconvex continuous bilinear program. While the cutting planes are obtained in a manner similar to Saxena, Bonami, and Lee [Math. Prog. 130: 359–413, 2011] and Fampa and Lee [J. Global … Read more

A Mixed-Integer PDE-Constrained Optimization Formulation for Electromagnetic Cloaking

We formulate a mixed-integer partial-differential equation constrained optimization problem for designing an electromagnetic cloak governed by the 2D Helmholtz equation with absorbing boundary conditions. Our formulation is an alternative to the topology optimization formulation of electromagnetic cloaking design. We extend the formulation to include uncertainty with respect to the angle of the incidence wave, and … Read more

A primal-dual modified log-barrier method for inequality constrained nonlinear optimization

We present a primal-dual modified log-barrier algorithm to solve inequality constrained nonlinear optimization problems. Basically, the algorithm is a Newton-like method applied to a perturbation of the optimality system that follows from a reformulation of the initial problem by introducing a modified log-barrier function to handle inequality constraints. The algorithm uses an outer/inner iteration scheme … Read more

Distributionally Robust Optimization under Decision-Dependent Ambiguity Set with an Application to Machine Scheduling

We introduce a new class of distributionally robust optimization problems under decision dependent ambiguity sets. In particular, as our ambiguity sets, we consider balls centered on a decision-dependent probability distribution. The balls are based on a class of earth mover’s distances that includes both the total variation distance and the Wasserstein metrics. We discuss the … Read more

Strong Evaluation Complexity Bounds for Arbitrary-Order Optimization of Nonconvex Nonsmooth Composite Functions

We introduce the concept of strong high-order approximate minimizers for nonconvex optimization problems. These apply in both standard smooth and composite non-smooth settings, and additionally allow convex or inexpensive constraints. An adaptive regularization algorithm is then proposed to find such approximate minimizers. Under suitable Lipschitz continuity assumptions, whenever the feasible set is convex, it is … Read more

Binary Optimal Control by Trust-Region Steepest Descent

We present a trust-region steepest descent method for dynamic optimal control problems with binary-valued integrable control functions. Our method interprets the control function as an indicator function of a measurable set and makes set-valued adjustments derived from the sublevel sets of a topological gradient function. By combining this type of update with a trust-region framework, … Read more