A Copositive Framework for Analysis of Hybrid Ising-Classical Algorithms

Recent years have seen significant advances in quantum/quantum-inspired technologies capable of approximately searching for the ground state of Ising spin Hamiltonians. The promise of leveraging such technologies to accelerate the solution of difficult optimization problems has spurred an increased interest in exploring methods to integrate Ising problems as part of their solution process, with existing approaches ranging from direct transcription to hybrid quantum-classical approaches rooted in existing optimization algorithms. While it is widely acknowledged that quantum computers should augment classical computers, rather than replace them entirely, comparatively little attention has been directed toward deriving analytical characterizations of their interactions. In this paper, we present a formal analysis of hybrid algorithms in the context of solving mixed-binary quadratic programs (MBQP) via Ising solvers. We show the exactness of a convex copositive reformulation of MBQPs, allowing the resulting reformulation to inherit the straightforward analysis of convex optimization. We propose to solve this reformulation with a hybrid quantum-classical cutting-plane algorithm. Using existing complexity results for convex cutting-plane algorithms, we deduce that the classical portion of this hybrid framework is guaranteed to be polynomial time. This suggests that when applied to NP-hard problems, the complexity of the solution is shifted onto the subroutine handled by the Ising solver

Strengthening Chvátal-Gomory cuts for the stable set problem

The stable set problem is a well-known NP-hard combinatorial optimization problem. As well as being hard to solve (or even approximate) in theory, it is often hard to solve in practice. The main difficulty is that upper bounds based on linear programming (LP) tend to be weak, whereas upper bounds based on semidefinite programming (SDP) take a long time to compute. We propose a new method to strengthen the LP-based upper bounds. The key idea is to take violated Chvátal-Gomory cuts and then strengthen their right-hand sides. Although the strengthening problem is itself NP-hard, it can be solved reasonably quickly in practice. As a result, the overall procedure proves to be capable of yielding competitive upper bounds in reasonable computing times

Proceedings of the XIII Global Optimization Workshop: GOW'16

[Excerpt] Preface: Past Global Optimization Workshop shave been held in Sopron (1985 and 1990), Szeged (WGO, 1995), Florence (GO’99, 1999), Hanmer Springs (Let’s GO, 2001), Santorini (Frontiers in GO, 2003), San José (Go’05, 2005), Mykonos (AGO’07, 2007), Skukuza (SAGO’08, 2008), Toulouse (TOGO’10, 2010), Natal (NAGO’12, 2012) and Málaga (MAGO’14, 2014) with the aim of stimulating discussion between senior and junior researchers on the topic of Global Optimization. In 2016, the XIII Global Optimization Workshop (GOW’16) takes place in Braga and is organized by three researchers from the University of Minho. Two of them belong to the Systems Engineering and Operational Research Group from the Algoritmi Research Centre and the other to the Statistics, Applied Probability and Operational Research Group from the Centre of Mathematics. The event received more than 50 submissions from 15 countries from Europe, South America and North America. We want to express our gratitude to the invited speaker Panos Pardalos for accepting the invitation and sharing his expertise, helping us to meet the workshop objectives. GOW’16 would not have been possible without the valuable contribution from the authors and the International Scientific Committee members. We thank you all. This proceedings book intends to present an overview of the topics that will be addressed in the workshop with the goal of contributing to interesting and fruitful discussions between the authors and participants. After the event, high quality papers can be submitted to a special issue of the Journal of Global Optimization dedicated to the workshop. [...

(Global) Optimization: Historical notes and recent developments

Recent developments in (Global) Optimization are surveyed in this paper. We collected and commented quite a large number of recent references which, in our opinion, well represent the vivacity, deepness, and width of scope of current computational approaches and theoretical results about nonconvex optimization problems. Before the presentation of the recent developments, which are subdivided into two parts related to heuristic and exact approaches, respectively, we briefly sketch the origin of the discipline and observe what, from the initial attempts, survived, what was not considered at all as well as a few approaches which have been recently rediscovered, mostly in connection with machine learning

Solution Techniques For Non-convex Optimization Problems

This thesis focuses on solution techniques for non-convex optimization problems. The first part of the dissertation presents a generalization of the completely positive reformulation of quadratically constrained quadratic programs (QCQPs) to polynomial optimization problems. We show that by explicitly handling the linear constraints in the formulation of the POP, one obtains a refinement of the condition introduced in Bai\u27s (2015) Thoerem on QCQPs, where the refined theorem only requires nonnegativity of polynomial constraints over the feasible set of the linear constraints. The second part of the thesis is concerned with globally solving non-convex quadratic programs (QPs) using integer programming techniques. More specifically, we reformulate non-convex QP as a mixed-integer linear problem (MILP) by incorporating the KKT condition of the QP to obtain a linear complementary problem, then use binary variables and big-M constraints to model the complementary constraints. We show how to impose bounds on the dual variables without eliminating all the (globally) optimal primal solutions; using some fundamental results on the solution of perturbed linear systems. The solution approach is implemented and labeled as quadprogIP, where computational results are presented in comparison with quadprogBB, BARON and CPLEX. The third part of the thesis involves the formulation and solution approach of a problem that arises from an on-demand aviation transportation network. A multi-commodity network flows (MCNF) model with side constraints is proposed to analyze and improve the efficiency of the on-demand aviation network, where the electric vertical-takeoff-and-landing (eVTOLs) transportation vehicles and passengers can be viewed as commodities, and routing them is equivalent to finding the optimal flow of each commodity through the network. The side constraints capture the decisions involved in the limited battery capacity for each eVTOL. We propose two heuristics that are efficient in generating integer feasible solutions that are feasible to the exponential number of battery side constraints. The last part of the thesis discusses a solution approach for copositive programs using linear semi-infinite optimization techniques. A copositive program can be reformulated as a linear semi-infinite program, which can be solved using the cutting plane approach, where each cutting plane is generated by solving a standard quadratic subproblem. Numerical results on QP-reformulated copositive programs are presented in comparison to the approximation hierarchy approach in Bundfuss (2009) and Yildirim (2012)