We propose a new approach to solving a wide class of optimization problems which fall into the broad framework of linear matrix inequalities and semidefinite programming. This approach based on randomization and cutting hyperplane ideology also covers robust statements of the problem. The proposed method is easy to implement, and it might be particularly useful in various aspects of functioning and applications of quantum computers.
We propose a new approach to solving a wide class of optimization problems which fall into the broad framework of linear matrix inequalities and semidefinite programming. This approach based on randomization and cutting hyperplane ideology also covers robust statements of the problem. The proposed method is easy to implement, and it might be particularly useful in various aspects of functioning and applications of quantum computers.
There is an increasing interest in quantum algorithms for optimization problems. Within convex optimization, interior-point methods and other recently proposed quantum algorithms are non-trivial to implement on noisy quantum devices. Here, we discuss how to utilize an alternative approach to convex optimization, in general, and semidefinite programming (SDP), in particular. This approach is based on a randomized variant of the cutting-plane method. We show how to leverage quantum speed-up of an eigensolver in speeding up an SDP solver utilizing the cutting-plane method. For the first time, we demonstrate a practical implementation of a randomized variant of the cutting-plane method for semidefinite programming on instances from SDPLIB, a well-known benchmark. Furthermore, we show that the RCP method is very robust to noise in the boundary oracle, which may make RCP suitable for use even on noisy quantum devices.
Optimization theory has been widely studied in academia and finds a large variety of applications in industry. The different optimization models in their discrete and/or continuous settings have catered to a rich source of research problems. Robust convex optimization is a branch of optimization theory in which the variables or parameters involved have a certain level of uncertainty. In this work, we consider the online robust optimization meta-algorithm by Ben-Tal et al. and show that for a large range of stochastic subgradients, this algorithm has the same guarantee as the original non-stochastic version. We develop a hybrid quantum-classical version of this algorithm and show that an at most quadratic improvement in terms of the dimension can be achieved. The speedup is due to the use of quantum state preparation, quantum norm estimation, and quantum multi-sampling. We apply our quantum meta-algorithm to examples such as robust linear programs and robust semidefinite programs and give applications of these robust optimization problems in finance and engineering.
This paper studies a fundamental problem in convex optimization, which is to solve semidefinite programming (SDP) with high accuracy. This paper follows from the existing robust SDP-based interior point method analysis due to [Huang, Jiang, Song, Tao and Zhang, FOCS 2022]. While, the previous work only provides an efficient implementation in the classical setting. This work provides a novel quantum implementation. We give a quantum second-order algorithm with high-accuracy in both the optimality and the feasibility of its output, and its running time depending on log(1/ε) on well-conditioned instances. Due to the limitation of quantum itself or first-order method, all the existing quantum SDP solvers either have polynomial error dependence or low-accuracy in the feasibility.
Multidimensional stochastic optimization plays an important role in analysis and control of many technical systems. To solve the challenging problems of multidimensional optimization, it was suggested to use the randomized algorithms of stochastic approximation with perturbed input which have simple forms and provide consistent estimates of the unknown parameters for observations under “almost arbitrary” noise. They are easily “incorporated” in the design of quantum devices to estimate gradient vector of a multi-variable function.
Many computer vision problems can be formulated as binary quadratic programs (BQPs). Two classic relaxation methods are widely used for solving BQPs, namely, spec-tral methods and semidefinite programming (SDP), each with their own advantages and disadvantages. Spectral relaxation is simple and easy to implement, but its bound is loose. Semidefinite relaxation has a tighter bound, but its computational complexity is high for large scale prob-lems. We present a new SDP formulation for BQPs, with two desirable properties. First, it has a similar relaxation bound to conventional SDP formulations. Second, com-pared with conventional SDP methods, the new SDP for-mulation leads to a significantly more efficient and scalable dual optimization approach, which has the same degree of complexity as spectral methods. Extensive experiments on various applications including clustering, image segmen-tation, co-segmentation and registration demonstrate the usefulness of our SDP formulation for solving large-scale BQPs. 1.
Solving optimization problems is a key task for which quantum computers could possibly provide a speedup over the best known classical algorithms. Particular classes of optimization problems including semidefinite programming (SDP) and linear programming (LP) have wide applicability in many domains of computer science, engineering, mathematics, and physics. Here we focus on semidefinite and linear programs for which the dimensions of the variables involved are exponentially large, so that standard classical SDP and LP solvers are not helpful for such large-scale problems. We propose the QSlack and CSlack methods for estimating their optimal values, respectively, which work by (1) introducing slack variables to transform inequality constraints to equality constraints, (2) transforming a constrained optimization to an unconstrained one via the penalty method, and (3) replacing the optimizations over all possible nonnegative variables by optimizations over parameterized quantum states and parameterized probability distributions. Under the assumption that the SDP and LP inputs are efficiently measurable observables, it follows that all terms in the resulting objective functions are efficiently estimable by either a quantum computer in the SDP case or a quantum or probabilistic computer in the LP case. Furthermore, by making use of SDP and LP duality theory, we prove that these methods provide a theoretical guarantee that, if one could find global optima of the objective functions, then the resulting values sandwich the true optimal values from both above and below. Finally, we showcase the QSlack and CSlack methods on a variety of example optimization problems and discuss details of our implementation, as well as the resulting performance. We find that our implementations of both the primal and dual for these problems approach the ground truth, typically achieving errors of the order 10-2.