Inference and Optimization for Engineering and Physical Systems

TL;DR

This thesis explores optimization techniques for complex systems like the Max Cut problem and Ising models, proposing heuristics and bi-level optimization methods applicable in fields such as physics, computer science, and pandemic modeling.

Contribution

It introduces new heuristics for improving semidefinite programming solutions and formulates a polynomial-time bi-level optimization approach for controlling Ising model ground states.

Findings

Semidefinite programming provides near-optimal Max Cut solutions.

Local heuristics can enhance rounding schemes.

Bi-level optimization is polynomial-time solvable with non-negative interactions.

Abstract

The central object of this PhD thesis is known under different names in the fields of computer science and statistical mechanics. In computer science, it is called the Maximum Cut problem, one of the famous twenty-one Karp's original NP-hard problems, while the same object from Physics is called the Ising Spin Glass model. This model of a rich structure often appears as a reduction or reformulation of real-world problems from computer science, physics and engineering. However, solving this model exactly (finding the maximal cut or the ground state) is likely to stay an intractable problem (unless $\textit{P} = \textit{NP}$) and requires the development of ad-hoc heuristics for every particular family of instances. One of the bright and beautiful connections between discrete and continuous optimization is a Semidefinite Programming-based rounding scheme for Maximum Cut. This procedure allows us to find a provably near-optimal solution; moreover, this method is conjectured to be the best possible in polynomial time. In the first two chapters of this thesis, we investigate local non-convex heuristics intended to improve the rounding scheme. In the last chapter of this thesis, we make one step further and aim to control the solution of the problem we wanted to solve in previous chapters. We formulate a bi-level optimization problem over the Ising model where we want to tweak the interactions as little as possible so that the ground state of the resulting Ising model satisfies the desired criteria. This kind of problem arises in pandemic modeling. We show that when the interactions are non-negative, our bi-level optimization is solvable in polynomial time using convex programming.