Quantum adiabatic evolution is perceived as useful for binary quadratic programming problems that are a priori unconstrained. For constrained problems, it is a common practice to relax linear equality constraints as penalty terms in the objective function. However, there has not yet been proposed a method for efficiently dealing with inequality constraints using the quantum adiabatic approach. In this paper, we give a method for solving the Lagrangian dual of a binary quadratic programming (BQP) problem in the presence of inequality constraints and employ this procedure within a branch-and-bound framework for constrained BQP (CBQP) problems.
We solve a multi-period portfolio optimization problem using D-Wave Systems' quantum annealer. We derive a formulation of the problem, discuss several possible integer encoding schemes, and present numerical examples that show high success rates. The formulation incorporates transaction costs (including permanent and temporary market impact), and, significantly, the solution does not require the inversion of a covariance matrix. The discrete multi-period portfolio optimization problem we solve is significantly harder than the continuous variable problem. We present insight into how results may be improved using suitable software enhancements, and why current quantum annealing technology limits the size of problem that can be successfully solved today. The formulation presented is specifically designed to be scalable, with the expectation that as quantum annealing technology improves, larger problems will be solvable using the same techniques.
A quantum annealing solver for the renowned job-shop scheduling problem (JSP) is presented in detail. After formulating the problem as a time-indexed quadratic unconstrained binary optimization problem, several pre-processing and graph embedding strategies are employed to compile optimally parametrized families of the JSP for scheduling instances of up to six jobs and six machines on the D-Wave Systems Vesuvius processor. Problem simplifications and partitioning algorithms, including variable pruning and running strategies that consider tailored binary searches, are discussed and the results from the processor are compared against state-of-the-art global-optimum solvers.
A quantum annealer heuristically minimizes quadratic unconstrained binary optimization (QUBO) problems, but is limited by the physical hardware in the size and density of the problems it can handle. We have developed a meta-heuristic solver that utilizes D-Wave Systems' quantum annealer (or any other QUBO problem optimizer) to solve larger or denser problems, by iteratively solving subproblems, while keeping the rest of the variables fixed. We present our algorithm, several variants, and the results for the optimization of standard QUBO problem instances from OR-Library of sizes 500 and 2500 as well as the Palubeckis instances of sizes 3000 to 7000. For practical use of the solver, we show the dependence of the time to best solution on the desired gap to the best known solution. In addition, we study the dependence of the gap and the time to best solution on the size of the problems solved by the underlying optimizer.
At 1QBit we have developed a method of measuring similarity between graphs with the aid of a quantum annealer. In contrast to conventional methods, our method is capable of identifying the reasons for determining that two arbitrary graphs are similar, and can illustrate how much each component of each graph contributes to the decision. Moreover, the method can be used to mine similar patterns in a group of graphs by solving subset matching problems. To validate our approach we apply our method in a biochemical scenario, classifying the toxicity properties of a library of molecules based on their similarity to labelled molecules. Benchmarking results show that this general-purpose similarity determination method can perform as accurately as the best-known classical solution while providing higher sensitivity or higher specificity, and maintaining the same predictive accuracy.
This research proposes a new approach for tackling cognitive radio asset allocation optimization problems. The cognitive radio optimization problem is generally an NP-hard mixed-integer programming problem due to its convoluted constraints. In contrast to conventional methods of using meta-heuristics and evolutionary algorithms, we implement non-linear constraints as polynomial penalty functions of binary variables and build a new objective function in a quadratic unconstrained binary optimization (QUBO) format.
This paper describes a general method developed by 1QBit for solving continuous optimization problems inspired by different types of simulated annealing and genetic algorithms. This method works under the assumption of the existence of a computation model with a Turing reduction of problems to either quadratic unconstrained binary optimization (QUBO) problems or to an Ising spin problem. The paper presents an application of this method to a mixed-integer optimization problem. This will demonstrate an interesting method of representing a cardinality-constrained optimization problem using analytic expressions, and the ability of the method to solve such mixed-integer optimization problems.
We describe how the Integer Optimization Toolbox approaches the problem of minimization of a (low-degree) polynomial over an integer lattice. In theory, the method illustrated here can be used for arbitrarily large polynomials with a large enough quantum annealing oracle.