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
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
Beginning in the 1960s, techniques from operations research began to be used to generate political districting plans. A classical example is the integer programming model of Hess et al. (Operations Research 13(6):998–1006, 1965). Due to the model’s compactness-seeking objective, it tends to generate contiguous or nearly-contiguous districts, although none of the model’s constraints explicitly impose … Read more
Motivated by modern regression applications, in this paper, we study the convexification of quadratic optimization problems with indicator variables and combinatorial constraints on the indicators. Unlike most of the previous work on convexification of sparse regression problems, we simultaneously consider the nonlinear objective, indicator variables, and combinatorial constraints. We prove that for a separable quadratic … Read more
Cutting planes accelerate branch-and-bound search primarily by cutting off fractional solutions of the linear programming (LP) relaxation, resulting in tighter bounds for pruning the search tree. Yet cutting planes can also reduce backtracking by excluding inconsistent partial assignments that occur in the course of branching. A partial assignment is inconsistent with a constraint set when … Read more
Deep neural networks have been successful in many predictive modeling tasks, such as image and language recognition, where large neural networks are often used to obtain good accuracy. Consequently, it is challenging to deploy these networks under limited computational resources, such as in mobile devices. In this work, we introduce an algorithm that removes units … Read more
In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization with \emph{non-Lipschitz-continuous} value functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient … Read more
Bilevel optimization problems have received a lot of attention in the last years and decades. Besides numerous theoretical developments there also evolved novel solution algorithms for mixed-integer linear bilevel problems and the most recent algorithms use branch-and-cut techniques from mixed-integer programming that are especially tailored for the bilevel context. In this paper, we consider MIQP-QP … Read more
We consider non-convex mixed-integer quadratic programs in which all variables are explicitly bounded. Many exact methods for such problems use additional variables, representing products of pairs of original variables. We study the convex hull of feasible solutions in this extended space. Some other approaches use bit representation to convert bounded integer variables into binary variables. … Read more
We present a convection-diffusion inverse problem that aims to identify an unknown number of sources and their locations. We model the sources using a binary function, and we show that the inverse problem can be formulated as a large-scale mixed-integer nonlinear optimization problem. We show empirically that current state-of-the-art mixed-integer solvers cannot solve this problem … Read more