LSP(n), the largest small polygon with n vertices, is defined as the polygon of unit diameter that has maximal area A(n). Finding the configuration LSP(n) and the corresponding A(n) for even values n >= 6 is a long-standing challenge that leads to an interesting class of nonlinear optimization problems. We present numerical solution estimates for … Read more
We present a model development framework and numerical solution approach to the general problem-class of packing convex objects into optimized convex containers. Specifically, here we discuss the problem of packing ovals (egg-shaped objects, defined here as generalized ellipses) into optimized regular polygons in R”. Our solution strategy is based on the use of embedded Lagrange … Read more
Signal estimation problems with smoothness and sparsity priors can be naturally modeled as quadratic optimization with L0-“norm” constraints. Since such problems are non-convex and hard-to-solve, the standard approach is, instead, to tackle their convex surrogates based on L1-norm relaxations. In this paper, we propose new iterative conic quadratic relaxations that exploit not only the L0-“norm” … Read more
In this article we introduce an inner parallel cutting plane method (IPCP) to compute good feasible points along with valid cutting planes for mixed-integer convex optimization problems. The method iteratively generates polyhedral outer approximations of an enlarged inner parallel set (EIPS) of the continuously relaxed feasible set. This EIPS possesses the crucial property that any … Read more
Semidefinite programming relaxations complement polyhedral relaxations for quadratic optimization, but global optimization solvers built on polyhedral relaxations cannot fully exploit this advantage. This paper develops linear outer-approximations of semidefinite constraints that can be effectively integrated into global solvers. The difference from previous work is that our proposed cuts are (i) sparser with respect to the … Read more
Primal-Dual Hybrid Gradient (PDHG) and Alternating Direction Method of Multipliers (ADMM) are two widely-used first-order optimization methods. They reduce a difficult problem to simple subproblems, so they are easy to implement and have many applications. As first-order methods, however, they are sensitive to problem conditions and can struggle to reach the desired accuracy. To improve … Read more
This paper describes and establishes the iteration-complexity of a doubly accelerated inexact proximal point (D-AIPP) method for solving the nonconvex composite minimization problem whose objective function is of the form f+h where f is a (possibly nonconvex) differentiable function whose gradient is Lipschitz continuous and h is a closed convex function with bounded domain. D-AIPP … Read more
We analyze a problem of dynamic logistics planning given uncertain demands for a multi-location production-inventory system with transportable modular production capacity. In such systems, production modules provide capacity, and can be moved from one location to another to produce stock and satisfy demand. We formulate a dynamic programming model for a planning problem that considers … Read more
In real life applications optimization problems with more than one objective function are often of interest. Next to handling multiple objective functions, another challenge is to deal with uncertainties concerning the realization of the decision variables. One approach to handle these uncertainties is to consider the objectives as set-valued functions. Hence, the image of one … Read more
A need for an optimal solution for a given mathematical model is well known and many solution approaches have been developed to identify efficiently an optimal solution in a given situation. For example, one such class of mathematical models with industrial applications have been classified as mathematical programming models (MPM). The main idea behind these … Read more