Abstract:In recent years the unconstrained binary quadratic program (UBQP) has grown in importance in the field of combinatorial optimization due to its application potential and its computational challenge. Research on UBQP has generated a wide range of solution techniques for this basic model that encompasses a rich collection of problem types. In this paper we survey the literature on this important model, providing an overview of the applications and solution methods.

S U P P L E M E N T A R Y NOTES~~~~i9 . ~~~~~~ w O R~~5 (C o ntinue or, reverse aid. if necessary and Id e n t i t y by bl ock number)~~ Utility functions Risk aversion Risk seeking LU Z~.~R A C T (Co, itlnue on teem,,. s ide II n e c e s s s r y end Idmo ' fr hi' blo ck number)Thirt y empiricall y-asses sed u t i lit y f u n c t i o n s on changes in we a lt h or retur on investment were examined for general features and susceptability to . fits by linear , power and exponential functions . Separate fits were made to below-* target data and above-target data. The usual target was the n-change point.The majority of below-target functions wore risk-seeking, the majority of above-target functions were risk-averse , and the most common composite shape was convex-concave , or risk-seeking in losses and risk-averse in gains.

The Quadratic Unconstrained Binary Optimization (QUBO) model has gained prominence in recent years with the discovery that it unifies a rich variety of combinatorial optimization problems. By its association with the Ising problem in physics, the QUBO model has emerged as an underpinning of the quantum computing area known as quantum annealing and Fujitsu's digital annealing, and has become a subject of study in neuromorphic computing. Through these connections, QUBO models lie at the heart of experimentation carried out with quantum computers developed by D-Wave Systems and neuromorphic computers developed by IBM. The consequences of these new computational technologies and their links to QUBO models are being explored in initiatives by organizations such as Google, Amazon and Lockheed Martin in the commercial realm and Los Alamos National Laboratory, Oak Ridge National Laboratory, Lawrence Livermore National Laboratory and NASA's Ames Research Center in the public sector. Computational experience is being amassed by both the classical and the quantum computing communities that highlights not only the potential of the QUBO model but also its effectiveness as an alternative to traditional modeling and solution methodologies.

Recent studies have demonstrated the effectiveness of applying adaptive memory tabu search procedures to combinatorial optimization problems. In this paper we describe the development and use of such an approach to solve binary quadratic programs. Computational experience is reported, showing that the approach optimally solves the most difficult problems reported in the literature. For challenging problems of limited size, which are capable of being approached by exact procedures, we find optimal solutions considerably faster than the best reported exact method. Moreover, we demonstrate that our approach is significantly more efficient and yields better solutions than the best heuristic method reported to date. Finally, we give outcomes for larger problems that are considerably more challenging than any currently reported in the literature.Integer Programming, Heuristics, Nonlinear Optimization

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