In this paper we investigate the dual of a Multistage Stochastic Linear Program (MSLP) to study two related questions for this class of problems. The first of these questions is the study of the optimal value of the problem as a function of the involved parameters. For this sensitivity analysis problem, we provide formulas for … Read more
We present two new algorithms for a class of single-objective non-linear optimization problems, the so-called Mixed Integer Linear minimum Multiplicative Programs (MIL-mMPs). This class of optimization problems has a desirable characteristic: a MIL-mMP can be viewed as a special case of the problem of optimization over the efficient set in multi-objective optimization. The proposed algorithms … Read more
A stochastic second-order trust region method is proposed, which can be viewed as a second-order extension of the trust-region-ish (TRish) algorithm proposed by Curtis et al. [INFORMS J. Optim. 1(3) 200–220, 2019]. In each iteration, a search direction is computed by (approximately) solving a trust region subproblem defined by stochastic gradient and Hessian estimates. The … Read more
Polynomial optimization problems over binary variables can be expressed as integer programs using a linearization with extra monomials in addition to those arising in the given polynomial. We characterize when such a linearization yields an integral relaxation polytope, generalizing work by Del Pia and Khajavirad (SIAM Journal on Optimization, 2018) and Buchheim, Crama and Rodríguez-Heck … Read more
We present a general numerical solution method for control problems with PDE-defined state variables over a finite set of binary or continuous control variables. We show empirically that a naive approach that applies a numerical discretization scheme to the PDEs (and if necessary a linearization scheme) to derive constraints for a mixed-integer linear program (MILP) … Read more
In this paper we conduct a study of both superiorization and optimization approaches for the reconstruction problem of superiorized/regularized solutions to underdetermined systems of linear equations with nonnegativity variable bounds. Specifically, we study a (smoothed) total variation regularized least-squares problem with nonnegativity constraints. We consider two approaches: (a) a superiorization approach that, in contrast to … Read more
A desirable property of integer formulations is to consist of few inequalities having small coefficients. We show that these targets are conflicting by proving the existence of knapsack sets that need exponentially many inequalities or exponentially large coefficients in any integer formulation. Moreover, we show that there exist undirected graphs such that (in a natural … Read more
In derivative-free and blackbox optimization, the objective function is often evaluated through the execution of a computer program seen as a blackbox. It can be noisy, in the sense that its outputs are contaminated by random errors. Sometimes, the source of these errors is identified and controllable, in the sense that it is possible to … Read more
This paper develops a novel limited-memory method to solve dynamic optimization problems. The memory requirements for such problems often present a major obstacle, particularly for problems with PDE constraints such as optimal flow control, full waveform inversion, and optical tomography. In these problems, PDE constraints uniquely determine the state of a physical system for a … Read more
This paper presents a novel price setting optimization problem for an energy retailer in the smart grid. In this framework the retailer buys energy from multiple generators via bilateral contracts, and sells it to a population of smart homes using Time-and-Level-of-Use prices (TLOU). TLOU is an energy price structure recently introduced in the literature, where … Read more