Thesis Title: Topics in Mixed Integer Nonlinear Optimization

Advisor:

Dr. Santanu S. Dey, School of Industrial and Systems Engineering, Georgia Institute of Technology

Thesis Committee:

Dr. Jean-Philippe P. Richard, Department of Industrial and Systems Engineering, University of Minnesota

Dr. Alejandro Toriello, School of Industrial and Systems Engineering, Georgia Institute of Technology

Dr. Nikolaos Sahinidis, School of Industrial and Systems Engineering, Georgia Institute of Technology

Dr. Diego Cifuentes, School of Industrial and Systems Engineering, Georgia Institute of Technology

Date and Time: Tuesday, July 5th, 2022, 03:00 PM (EST)

Location: Groseclose 404 or meeting link below

Meeting Link: https://gatech.zoom.us/j/93882866413?pwd=U1Y2NjZTNzlRNnk0NHpiWTg3MXZrdz09

Meeting ID: 938 8286 6413

Passcode: 388052

Abstract:

Mixed integer nonlinear optimization has many applications ranging from machine learning to power systems. However, these problems are very challenging to solve to global optimality due to the inherent non-convexity. This typically leads the problem to be NP-hard. Moreover, in many applications, there are time and resource limitations for solving real-world problems, and the sheer size of real instances can make solving them challenging. In this thesis, we focus on important elements of nonconvex optimization - including mixed integer linear programming and nonlinear programming, where both theoretical analyses and computational experiments are presented.

In Chapter 1 we look at Mixed Integer Quadratic Programming (MIQP), the problem of minimizing a convex quadratic function over mixed integer points in a rational polyhedron. We utilize the augmented Lagrangian dual (ALD), which augments the usual Lagrangian dual with a weighted nonlinear penalty on the dualized constraints. We first prove that ALD will reach a zero duality gap asymptotically as the weight on the penalty goes to infinity under some mild conditions on the penalty function. We next show that a finite penalty weight is enough for a zero gap when we use any norm as the penalty function. Finally, we prove a polynomial bound on the weight on the penalty term to obtain a zero gap.

In Chapter 2 we apply the technique of lifting to bilinear programming, a special case of quadratic constrained quadratic programming. We first show that, for sets described by one bilinear constraint together with bounds, it is always possible to sequentially lift a seed inequality. To reduce computational burden, we develop a framework based on subadditive approximations of lifting functions that permits sequence-independent lifting of seed inequalities for separable bilinear sets. We then study a separable bilinear set where the coefficients form a minimal cover with respect to the right-hand-side. For this set, we introduce a bilinear cover inequality, which is second-order cone representable. We study the lifting function of the bilinear cover inequality and lift fixed variable pairs in closed-form, thus deriving a lifted bilinear cover inequality that is valid for general separable bilinear sets with box constraints.

In Chapter 3 we continue our research around separable bilinear programming. We begin by designing a simple randomized separation heuristic for these inequalities. In our computational experiments, we separate many rounds of these inequalities starting from the McCormick relaxation of bilinear instances where each constraint is a separable bilinear constraint set. Our main result is to demonstrate that there is a significant improvement in the performance of a state-of-the-art global solver in terms of the gap closed, when these inequalities are added at the root node compared to when these inequalities are not added.

In Chapter 4 we look at Mixed Integer Linear Programming (MILP) that arises in operational applications. Many routinely-solved MILPs are extremely challenging not only from a worst-case complexity perspective, but also because of the necessity to obtain good solutions within limited time. An example is the Security-Constrained Unit Commitment (SCUC) problem, solved daily to clear the day-ahead electricity markets. We develop ML-based methods for improving branch-and-bound variable selection rules that exploit key features of such operational problems: similar decisions are generated within the same day and across different days, based on the same power network. Utilizing similarity between instances and within an instance, we build one separate ML model per variable or per group of similar variables for learning to predict the strong branching score. The approach is able to produce branch-and-bound trees which gap closed only slightly worse than that of trees obtained by strong branching, while it outperforms previous machine learning schemes.