Quantum computing for solving nonlinear optimization problems

Dilnoz Muhamediyeva, Rustam Raximov, Shavkat Urokov

Ashfaq Iftakher, M. M. Faruque Hasan

Computer-aided Molecular and Process Design (CAMPD) is an equation-oriented multi-scale decision making framework for designing both materials (molecules) and processes for separation, reaction, and reactive separation whenever material choice significantly impacts process performance. The inherent nonlinearity and nonconvexity in CAMPD optimization models, introduced through the property and process models, pose challenges to state-of-the-art solvers. Recently, quantum computing (QC) has shown promise for solving complex optimization problems, especially those involving discrete decisions. This motivates us to explore the potential usage of quantum optimization techniques for solving CAMPD problems. We have developed a technique for directly solving a class of mixed integer nonlinear programs using QC. Our approach represents both continuous and integer design decisions by a set of binary variables through encoding schemes. This transformation allows to reformulate certain types of CAMPD problems into Quadratic Unconstrained Binary Optimization (QUBO) models that can be directly solved using quantum annealing techniques. We illustrate this technique for the selection of optimal ionic liquids (IL) and the configuration of a reactor-separator process network. We also discuss several challenges that are associated with quantum optimization when solving large scale CAMPD problems.

Dawid Ewald

The welded beam design problem represents a real-world engineering challenge in structural optimization. The objective is to determine the optimal dimensions of a steel beam and weld length to minimize cost while satisfying constraints related to shear stress (τ), bending stress (σ), critical buckling load (Pc), end deflection (δ), and side constraints. The structural analysis of this problem involves the following four design variables: weld height (x1), weld length (x2), beam thickness (x3), and beam width (x4), which are commonly denoted in structural engineering as h,l,t,b respectively. The structural formulation of this problem leads to a nonlinear objective function, which is subject to five nonlinear and two linear inequality constraints. The optimal solution lies on the boundary of the feasible region, with a very small feasible-to-search-space ratio, making it a highly challenging problem for classical optimization algorithms. This paper explores the application of quantum computing to solve the welded beam optimization problem, utilizing the unique properties of quantum computers for constrained optimization in engineering problems. Specifically, we employ the D-Wave quantum computing system, which utilizes quantum annealing and is particularly well-suited for solving constrained optimization problems. The study presents a detailed formulation of the problem in a format compatible with the D-Wave system, ensuring the efficient encoding of constraints and objective functions. Furthermore, we analyze the performance of quantum computing in solving this problem and compare the obtained results with classical optimization methods. The effectiveness of quantum computing is evaluated in terms of computational efficiency, accuracy, and its ability to navigate complex, constrained search spaces. This research highlights the potential of quantum algorithms in tackling real-world engineering optimization problems and discusses the challenges and limitations of current quantum hardware in solving practical industrial application issues.

Mitsuki Ito, Y. Hirata, Moe Shimada +2 more

Mengyu Guo, Shaowen Wang

Nimna Suharshani, Awansika Nimuthumana, Geethika Prabhath +4 more

Quantum annealing is a recent advancement in the field of optimization metaheuristics. Quantum annealers such as the D-Wave machines are primarily designed for solving discrete optimization problems. Solving nonlinear continuous optimization problems is also a very challenging task, in particular when nonconvex functions and multiple local optima are involved; nevertheless, convex problems can be solved efficiently using standard classical methods. It has not been explored in sufficient detail how quantum annealing could help in solving continuous optimization problems. In this work, we develop a prototype quantum-inspired solver for nonlinear continuous optimization problems. The solver is hybrid, as it has two pathways - classical and quantum-inspired. Given a continuous optimization problem via the web-based graphical interface that enables users to define and edit optimization problems, our solver computes a nonconvexity index for the problem, which determines the solution pathway. Problems with low nonconvexity indices are sent through the classical pathway and solved by standard separable programming techniques, while the others are transformed into mixed integer linear programming (MILP) problems via separable reformulations, which are restated as quadratic unconstrained binary optimization (QUBO) problems - the problem format compatible for quantum annealers, and eventually sent through the quantum pathway using D-Wave’s Neal sampler.