Second-Order Cone Relaxations for Binary Quadratic Polynomial Programs

This paper is concerned with binary quadratic programs (BQPs), which are among the most well-studied classes of nonlinear integer optimization problems because of their wide variety of applications. While a number of different solution approaches have been proposed for tackling BQPs, practitioners need techniques that are both efficient and easy to implement. We revisit two of the most widely used linearization strategies for BQPs and examine the effectiveness of enhancements to these formulations that have been suggested in the literature. We perform a detailed large-scale computational study over five different classes of BQPs to compare these two linearizations with a more recent linear reformulation and direct submission of the nonlinear integer program to an optimization solver. The goal is to provide practitioners with guidance on how to best approach solving BQPs in an effective and easily implemented manner.

Quadratic programming (QP) is a family of methods, techniques, and algorithms that can be used to minimize quadratic objective functions subject to linear constraints. QP shares many combinatorial features with linear programming (LP) and it is often used as the basis of constrained nonlinear programming. An important branch of QP is convex QP where the objective function is a convex quadratic function.

The quadratic shortest path problem (QSPP) has a lot of applications in real-life problems. This thesis consists of four essays about the QSPP concerning its theory and computation, where linear and semidefinite programming were used as a major tool for solving and understanding the problem. In the second chapter, we investigate the computational complexity of the problem as well as the polynomial-time solvable special cases. In the third and fourth chapters, we show that the linearization problem of the quadratic shortest path problem on directed acyclic graphs can be solved efficiently. By using the linearization problem of binary quadratic problems as a unifying tool, the dominance between the Generalized Gilmore-Lawler bound and the first level RLT bound is also established. We also propose a new class of relaxation for binary quadratic optimization problems based on its linearization problem. In the last chapter, we derive several semidefinite programming relaxations with increasing complexity for the quadratic shortest path problem. We implement the alternating direction method of multipliers to solve our strongest semidefinite programming relaxation, and the obtained bound is currently the best one solved in the literature.

A linearization method for mixed 0–1 polynomial programs

Semidefinite relaxation method for unified near-Field and far-Field localization by AOA

In this paper, we propose a bounded degree hierarchy of both primal and dual conic programming relaxations involving both semi-definite and second-order cone constraints for solving a nonconvex polynomial optimization problem with a bounded feasible set. This hierarchy makes use of some key aspects of the convergent linear programming relaxations of polynomial optimization problems (Lasserre in Moments, positive polynomials and their applications, World Scientific, Singapore, 2010) associated with Krivine–Stengle’s certificate of positivity in real algebraic geometry and some advantages of the scaled diagonally dominant sum of squares (SDSOS) polynomials (Ahmadi and Hall in Math Oper Res, 2019. https://doi.org/10.1287/moor.2018.0962; Ahmadi and Majumdar in SIAM J Appl Algebra Geom 3:193–230, 2019). We show that the values of both primal and dual relaxations converge to the global optimal value of the original polynomial optimization problem under some technical assumptions. Our hierarchy, which extends the so-called bounded degree Lasserre hierarchy (Lasserre et al. in Eur J Comput Optim 5:87–117, 2017), has a useful feature that the size and the number of the semi-definite and second-order cone constraints of the relaxations are fixed and independent of the step or level of the approximation in the hierarchy. As a special case, we provide a convergent bounded degree second-order cone programming (SOCP) hierarchy for solving polynomial optimization problems. We then present finite convergence at step one of the SOCP hierarchy for classes of polynomial optimization problems. This includes one-step convergence for a new class of first-order SDSOS-convex polynomial programs. In this case, we also show how a global solution is recovered from the level one SOCP relaxation. We finally derive a corresponding convergent conic linear programming hierarchy for conic-convex semi-algebraic programs. Whenever the semi-algebraic set of the conic-convex program is described by concave polynomial inequalities, we show further that the values of the relaxation problems converge to the common value of the convex program and its Lagrangian dual under a constraint qualification.

The dataset contains instances for the cardinality constrained quadratic knapsack problem. These are used to test decomposition methods for Binary Quadratic Programs. Full details are given in the corresponding paper Dantzig-Wolfe Reformulations for Binary Quadratic Problems . The dataset contains three sets of instances - qcr_instances: base instance, together with the optimal quadratic convex reformulation (QCR) multipliers found by solving an associated semidefinite program - mqcr_instances: base instance, together with the optimal improved convex 0-1 quadratic program reformulation (MQCR) multipliers found by solving an associated semidefinite program - convexity_analysis: instances in which the objective function quadratic cost matrix has a given number of positive eigenvalues.

Binary quadratic programming (BQP) is a typical integer programming problem widely applied in the field of signal processing, economy, management and engineering. However, it's NP-hard and lacks efficient algorithms. Due to this reason, in this paper, a novel polynomial algorithm to linearly constrained binary quadratic programming problems with Q being a five-diagonal matrix is focused by combining the basic algorithm proposed in [1], [2], [3] and the dynamic programming method. We first briefly deduce the basic algorithm. Then, the algorithm is proposed to solve this special problem. In addition, a specific example is presented to illustrate the new algorithm. Lastly, we demonstrate its polynomial feature as well as its high efficiency.

New Advances in Convex Optimization and Control Applications

Recent research shows that non-convex AC OPF problem can be recast as a convex Semidefinite (SD) problem or Second Order Cone Programming (SOCP) problem. This paper presents a stochastic SOCP OPF (SSOCP-OPF) model for power systems connected to the wind farms. This is performed by reformulating the original non-convex OPF problem using more practical parameters of the power system. Finally, we obtain a convex optimization problem through some well-known approximations and an exact relaxation incorporating the stochastic nature of wind power. One of the advantage of convex SOCP problems, which are a general form of linear problems, is that they can be efficiently solved through Interior Point Methods (IPMs). The proposed OPF model takes advantage of both DC-OPF models (solution efficiency) and full AC-OPF models (solution accuracy). As an application of SSOCP-OPF model, we study the impact of wind power uncertainty on the transmission loss in the power systems. To evaluate the proposed stochastic model modified IEEE 30-bus test system is used. The optimization problem is coded in GAMS platform and solved using its embedded interior point optimizer MOSEK.

In this paper, we propose a new multiuser detector that is based on the explicit estimation of Lagrange multipliers. The optimum multiuser detector, proposed by S. Verdu in 1986, is a binary quadratic problem. It belongs to NP hard problems and can not be used currently. In this paper, the binary quadratic problem is changed into a quadratic optimization problem with equation constraints, and then written in Lagrange form, the relationship between the Lagrange multipliers, user information codes and channel noise is derived. According to different methods of estimating Lagrange multipliers, two existing multiuser detectors are given: decorrelation and minimum mean-squared error (MMSE) detectors. According to the special features of the multiuser detection problem, a new method of estimating Lagrange multipliers is given in this paper and then a new multiuser detector is obtained. The numerical results showed that the bit-error-ratio (BER) of the new detector is close to that of the optimum multiuser detector and less than that of sub-optimum multiuser detection.

A column generation approach for the unconstrained binary quadratic programming problem

We study support vector machines (SVM) for which the kernel matrix is not specified exactly and it is only known to belong to a given uncertainty set. We consider uncertainties that arise from two sources: (i) data measurement uncertainty, which stems from the statistical errors of input samples; (ii) kernel combination uncertainty, which stems from the weight of individual kernel that needs to be optimized in multiple kernel learning (MKL) problem. Much work has been studied, such as uncertainty sets that allow the corresponding SVMs to be reformulated as semi-definite programs (SDPs), which is very computationally expensive however. Our focus in this paper is to identify uncertainty sets that allow the corresponding SVMs to be reformulated as second-order cone programs (SOCPs), since both the worst case complexity and practical computational effort required to solve SOCPs is at least an order of magnitude less than that needed to solve SDPs of comparable size. In the main part of the paper we propose four uncertainty sets that meet this criterion. Experimental results are presented to confirm the validity of these SOCP reformulations.

Solving optimization problems with sparse or low-rank optimal solutions has been an important topic since the recent emergence of compressed sensing and its matrix extensions such as the matrix rank minimization and robust principal component analysis problems. Compressed sensing enables one to recover a signal or image with fewer observations than the “length” of the signal or image, and thus provides potential breakthroughs in applications where data acquisition is costly. However, the potential impact of compressed sensing cannot be realized without efficient optimization algorithms that can handle extremely large-scale and dense data from real applications. Although the convex relaxations of these problems can be reformulated as either linear programming, second-order cone programming or semidefinite programming problems, the standard methods for solving these relaxations are not applicable because the problems are usually of huge size and contain dense data. In this dissertation, we give efficient algorithms for solving these “sparse” optimization problems and analyze the convergence and iteration complexity properties of these algorithms. Chapter 2 presents algorithms for solving the linearly constrained matrix rank minimization problem. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast and solved as a semidefinite programming problem, such an approach is computationally expensive when the matrices are large. In Chapter 2, we propose fixed-point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems. Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 × 1000 matrices of rank 50 with a relative error of 10−5 in about 3 minutes by sampling only 20 percent of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms. In Chapter 3, we study the convergence/recoverability properties of the fixed-point continuation algorithm and its variants for matrix rank minimization. Heuristics for determining the rank of the matrix when its true rank is not known are also proposed. Some of these algorithms are closely related to greedy algorithms in compressed sensing. Numerical results for these algorithms for solving linearly constrained matrix rank minimization problems are reported. Chapters 4 and 5 considers alternating direction type methods for solving composite convex optimization problems. We present in Chapter 4 alternating linearization algorithms that are based on an alternating direction augmented Lagrangian approach for minimizing the sum of two convex functions. Our basic methods require at most O(1/e) iterations to obtain an e-optimal solution, while our accelerated (i.e., fast) versions require at most O(1/ 3 ) iterations, with little change in the computational effort required at each iteration. For more general problem, i.e., minimizing the sum of K convex functions, we propose multiple-splitting algorithms for solving them. We propose both basic and accelerated algorithms with O(1/e) and O(1/ 3 ) iteration complexity bounds for obtaining an e-optimal solution. To the best of our knowledge, the complexity results presented in these two chapters are the first ones of this type that have been given for splitting and alternating direction type methods. Numerical results on various applications in sparse and low-rank optimization, including compressed sensing, matrix completion, image deblurring, robust principal component analysis, are reported to demonstrate the efficiency of our methods.

Genomic selection (GS) in plant breeding aims to identify individuals with superior genetic merit while maintaining genetic diversity within populations. In plant breeding, considering multiple traits simultaneously makes optimizing selection complex, especially under genetic relatedness constraints. In this study, we propose a binary quadratic programming framework for constructing a multi-trait selection index that maximizes genetic gain while minimizing average pairwise relatedness appropriate for identifying superior candidates for advancement in the breeding pipeline. The approach combines estimated breeding values (EBVs) across multiple traits by applying trait-specific economic weights, while simultaneously accounting for coancestry through the genomic relationship matrix. By formulating the selection problem as a constrained Quadratic Programing Multi-trait Selection Index (QPMSI), our method enables the identification of a fixed number of candidate individuals that jointly optimize selection index values and control genetic relatedness. We evaluated the performance of the proposed method using five real genomic datasets and demonstrated that it provides a more effective balance between selection response and control of genetic relatedness than the Linear Programming Multi-trait Selection Index (LPMSI). In particular, the QPMSI consistently outperformed the LPMSI in terms of the MV metric (gain-to-degree of relatedness ratio), achieving improvements of at least 53.8%. This framework offers a practical and computationally efficient tool for sustainable breeding strategies in multi-trait selection contexts.