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.

We prove that the bundle method for nonsmooth optimization achieves solution accuracy ε in at most O ln(1/ε)/ε iterations, if the function is strongly convex. The result is true for the versions of the method with multiple cuts and with cut aggregation.

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Quantum Bridge Analytics relates to methods and systems for hybrid classicalquantum computing, and is devoted to developing tools for bridging classical and quantum computing to gain the benefits of their alliance in the present and enable enhanced practical application of quantum computing in the future.This is the second of a two-part tutorial that surveys key elements of Quantum Bridge Analytics and its applications. Part I focused on the Quadratic Unconstrained Binary Optimization (QUBO) model which is presently the most widely applied optimization model in the quantum computing area, and which unifies a rich variety of combinatorial optimization problems. Part II (the present paper) introduces the domain of QUBO-Plus models that enables a larger range of problems to be handled effectively. After illustrating the scope of these QUBO-Plus models with examples, we give special attention to an important instance of these models called the Asset Exchange Problem (AEP). Solutions to the AEP enable market players to identify exchanges of assets that benefit all participants. Such exchanges are generated by a combination of two optimization technologies for this class of QUBO-Plus models, one grounded in network optimization and one based on a new metaheuristic optimization approach called combinatorial chaining. This combination opens the door to expanding the links to quantum computing applications established by QUBO models through the Quantum Bridge Analytics perspective. We show how the modeling and solution capability for the AEP instance of QUBO-Plus models provides a framework for solving a broad range of problems arising in financial, industrial, scientific, and social settings.

We consider the problem of minimizing a sum of several convex non-smooth functions. We introduce a new algorithm called the selective linearization method, which iteratively linearizes all but one of the functions and employs simple proximal steps. The algorithm is a form of multiple operator splitting in which the order of processing partial functions is not fixed, but rather determined in the course of calculations. Global convergence is proved and estimates of the convergence rate are derived. Specifically, the number of iterations needed to achieve solution accuracy ε is of order O ln(1/ε)/ε . We also illustrate the operation of the algorithm on structured regularization problems.