In this paper, we consider a binary quadratic program (BQP) with random objective coefficients. Given only information on the marginal distributions of the objective coefficients, we propose a tight bound on the expected optimal value of the random BQP. We show that the complexity of computing this bound does not increase substantially with respect to the complexity of solving the corresponding deterministic BQP. For the quadratic unconstrained binary optimization (QUBO) problem with nonnegative off-diagonal random entries, the bound is shown to be computable in polynomial time. We generalize the asymptotic bound for the random quadratic assignment problem from independent random variables to dependent random variables and propose a new closed-form bound on the expected optimal value of the quadratic k-cluster problem. We also provide polynomial time computable upper bounds on the expected optimal value for the NP-hard instances using the linear and semidefinite programming relaxation of the deterministic BQP. The semidefinite programming bound of the expected optimal value of the random MAX-CUT problem inherits the approximation ratio of 0.878 from the semidefinite relaxation of the deterministic MAX-CUT problem. Computational experiments on random QUBO problems and random quadratic knapsack problems provide evidence of the quality of the bounds. Overall, our results indicate that the new bound on the expected optimal value of the random BQP is attractive since it exploits many of the nice results and formulations of the deterministic BQP and is valid under limited distributional information.
In this paper, we consider a binary quadratic program (BQP) with random objective coefficients. Given only information on the marginal distributions of the objective coefficients, we propose a tight bound on the expected optimal value of the random BQP. We show that the complexity of computing this bound does not increase substantially with respect to the complexity of solving the corresponding deterministic BQP. For the quadratic unconstrained binary optimization (QUBO) problem with nonnegative off-diagonal random entries, the bound is shown to be computable in polynomial time. We generalize the asymptotic bound for the random quadratic assignment problem from independent random variables to dependent random variables and propose a new closed-form bound on the expected optimal value of the quadratic k-cluster problem. We also provide polynomial time computable upper bounds on the expected optimal value for the NP-hard instances using the linear and semidefinite programming relaxation of the deterministic BQP. The semidefinite programming bound of the expected optimal value of the random MAX-CUT problem inherits the approximation ratio of 0.878 from the semidefinite relaxation of the deterministic MAX-CUT problem. Computational experiments on random QUBO problems and random quadratic knapsack problems provide evidence of the quality of the bounds. Overall, our results indicate that the new bound on the expected optimal value of the random BQP is attractive since it exploits many of the nice results and formulations of the deterministic BQP and is valid under limited distributional information.
Quadratic optimization problems with complex-valued decision variables, in short called complex quadratic optimization problems, find many applications in engineering. In this dissertation, we study several instructive models of complex quadratic optimization, as well as its applications in combinatorial optimization. The tool that we use is a combination of semidefinite programming (SDP) relaxation and randomization technique, which has been well exploited in the last decade. Since most of the optimization models are NP-hard in nature, we shall design polynomial time approximation algorithms for a general model, or polynomial time exact algorithms for some restricted instances of a general model.
To enable the analysis, we first develop two basic theoretical results: one is the probability formula for a bivariate complex normal distribution vector to be in a prescribed angular region, and the other one is the rank-one decomposition theorem for complex positive semidefinite matrices. The probability formula enables us to compute the expected value of a randomized (with a specific rounding rule) solution based on an optimal solution of the SDP relaxation problem, while the rank-one decomposition theorem provides a new proof of the complex S -lemma and leads to novel deterministic rounding procedures.
With the above results in hand, we then investigate the models and applications of complex quadratic optimization via semidefinite programming in detail. We present an approximation algorithm for a convex quadratic maximization problem with discrete complex decision variables, where the approximation analysis is based on the probability formula. Besides, an approximation algorithm is proposed for a non-convex quadratic maximization problem with discrete complex decision variables. Then we study a limit of the model, i.e., a quadratic maximization problem with continuous unit module decision variables. The problem is shown to be strongly NP-hard. Approximation algorithms are described for the problem, including both convex case and non-convex case. Furthermore, if the objective matrix has a sign structure, then a stronger approximation result is shown to hold. In addition, we use the complex matrix decomposition theorem to solve complex quadratically constrained complex quadratic programming. We consider several interesting cases for which the corresponding SDP relaxation admits no gap with the true optimal value, and consequently, this yields polynomial time procedures for solving those special cases of complex quadratic optimization.
Finally, as combinatorial applications of complex quadratic optimization; we consider Max 3-Cut with fixed nodes constraints, Max 3-Dicut with weight constraints, Max 3-XOR, and so on, and present corresponding bounds on the approximation ratios.
It has recently been shown (Burer, Math. Program Ser. A 120:479-495, 2009) that a large class of NP-hard nonconvex quadratic programming problems can be modeled as so called completely positive programming problems, which are convex but still NP-hard in general. A basic tractable relaxation is gotten by doubly nonnegative relaxation, resulting in a doubly nonnegative programming. In this paper, we prove that doubly nonnegative relaxation for binary quadratic programming (BQP) problem is equivalent to a tighter semidifinite relaxation for it. When problem (BQP) reduces to max-cut (MC) problem, doubly nonnegative relaxation for it is equivalent to the standard semidifinite relaxation. Furthermore, some compared numerical results are reported.
Abstract Quadratic Unconstrained Binary Optimization (QUBO or UBQP) is concerned with maximizing/minimizing the quadratic form H(J,η)=W∑i,jJi,jηiηj H ( J , η ) = W ∑ i , j J i , j η i η j with J a matrix of coefficients, η∈{0,1}N η ∈ { 0 , 1 } N and W a normalizing constant. In the statistical mechanics literature, QUBO is a lattice gas counterpart to the (generalized) Sherrington–Kirkpatrick spin glass model. Finding the optima of H is an NP-hard problem. Several problems in combinatorial optimization and data analysis can be mapped to QUBO in a straightforward manner. In the combinatorial optimization literature, random instances of QUBO are often used to test the effectiveness of heuristic algorithms. Here we consider QUBO with random independent coefficients and show that if the Ji,j J i , j ’s have zero mean and finite variance then, after proper normalization, the minimum and maximum per particle of H do not depend on the details of the distribution of the couplings and are concentrated around their expected values. Further, with the help of numerical simulations, we study the minimum and maximum of the objective function and provide some insight into the structure of the minimizer and the maximizer of H. We argue that also this structure is rather robust. Our findings hold also in the diluted case where each of the Ji,j J i , j ’s is allowed to be zero with probability going to 1 as N→∞ N → ∞ in a suitable way.
In computer vision, many problems can be formulated as binary quadratic programs (BQPs), which are in general NP hard. Finding a solution when the problem is of large size to be of practical interest typically requires relaxation. Semidefinite relaxation usually yields tight bounds, but its computational complexity is high. In this work, we present a semidefinite programming (SDP) formulation for BQPs, with two desirable properties. First, it produces similar bounds to the standard SDP formulation. Second, compared with the conventional SDP formulation, the proposed SDP formulation leads to a considerably more efficient and scalable dual optimization approach. We then propose two solvers, namely, quasi-Newton and smoothing Newton methods, for the simplified dual problem. Both of them are significantly more efficient than standard interior-point methods. Empirically the smoothing Newton solver is faster than the quasi-Newton solver for dense or medium-sized problems, while the quasi-Newton solver is preferable for large sparse/structured problems.
Binary programs with a quadratic objective function are NP-hard in general, even if the linear optimization problem over the same feasible set is tractable. In this paper, we address such problems by computing quadratic global underestimators of the objective function that are separable but not necessarily convex. Exploiting the binary constraint on the variables, a minimizer of the separable underestimator over the feasible set can be computed by solving an appropriate linear minimization problem over the same feasible set. Embedding the resulting lower bounds into a branch-and-bound framework, we obtain an exact algorithm for the original quadratic binary program. The main practical challenge is the fast computation of an appropriate underestimator, which in our approach reduces to solving a series of semidefinite programs. We exploit the special structure of the resulting problems to obtain a tailored coordinate-descent method for their solution. Our extensive experimental results on various quadratic combinatorial optimization problems show that our approach outperforms both CPLEX and the related QCR method as well as the SDP-based software BiqCrunch on instances of the quadratic shortest path problem and the quadratic assignment problem.
In this paper we study the approximation algorithms for a class of discrete quadratic optimization problems in the Hermitian complex form. A special case of the problem that we study corresponds to the max-3-cut model used in a recent paper of Goemans and Williamson J. Comput. System Sci., 68 (2004), pp. 442-470]. We first develop a closed-formformula to compute the probability of a complex-valued normally distributed bivariate random vector to be in a given angular region. This formula allows us to compute the expected value of a randomized (with a specific rounding rule) solution based on the optimal solution of the complex semidefinite programming relaxation problem. In particular, we present an [m2(1−cosm2π)/8π]-approximation algorithm, and then study the limit of that model, in which the problem remains NP-hard. We show that if the objective is to maximize a positive semidefinite Hermitian form, then the randomization-rounding procedure guarantees a worst-case performance ratio of π/4≈0.7854, which is better than the ratio of 2/π≈0.6366 for its counterpart in the real case due to Nesterov. Furthermore, if the objective matrix is real-valued positive semidefinite with nonpositive off-diagonal elements, then the performance ratio improves to 0.9349.