In 2004 Newman [43] suggested a semidefinite programming relaxation for the Linear Ordering Problem (LOP) that is related to the semidefinite program used in the Goemans-Williamson algorithm to approximate the Max Cut...
详细信息
In 2004 Newman [43] suggested a semidefinite programming relaxation for the Linear Ordering Problem (LOP) that is related to the semidefinite program used in the Goemans-Williamson algorithm to approximate the Max Cut problem (Goemans and Williamson, 1995). Her model is based on the observation that linear orderings can be fully described by a series of cuts. Newman (2004) [43] shows that her relaxation seems better suited for designing polynomial-time approximation algorithms for the (LOP) than the widely-studied standard polyhedral linear relaxations. In this paper we strengthen the relaxation proposed by Newman (2004) [43] and conduct a polyhedral study of the corresponding polytope. Furthermore we relate the relaxation to other linear and semidefinite relaxations for the (LOP) and for the Traveling Salesman Problem and elaborate on its connection to the Max Cut problem. (C) 2016 Elsevier B.V. All rights reserved.
This paper addresses truss topology optimization taking into account robustness to uncertainty in the truss geometry. Specifically, the locations of nodes are assumed not to be known precisely and the compliance in th...
详细信息
This paper addresses truss topology optimization taking into account robustness to uncertainty in the truss geometry. Specifically, the locations of nodes are assumed not to be known precisely and the compliance in the worst case is attempted to be minimized. We formulate a semidefinite programming problem that serves as a safe approximation of this robust optimization problem. That is, any feasible solution of the presented semidefinite programming problem satisfies the constraints of the original robust optimization problem. Since a semidefinite programming problem can be solved efficiently with a primal-dual interior-point method, we can find a robust truss design efficiently with the proposed semidefinite programming approach. A notable property of the proposed approach is that the obtained truss is guaranteed to be stable. Numerical experiments are performed to illustrate that the optimal truss topology depends on the magnitude of uncertainty.
A wide variety of problems in global optimization, combinatorial optimization, as well as systems and control theory can be solved by using linear and semidefinite programming. Sometimes, due to the use of floating po...
详细信息
A wide variety of problems in global optimization, combinatorial optimization, as well as systems and control theory can be solved by using linear and semidefinite programming. Sometimes, due to the use of floating point arithmetic in combination with ill-conditioning and degeneracy, erroneous results may be produced. The purpose of this article is to show how rigorous error bounds for the optimal value can be computed by carefully postprocessing the output of a linear or semidefinite programming solver. It turns out that in many cases the computational costs for postprocessing are small compared to the effort required by the solver. Numerical results are presented including problems from the SDPLIB and the NETLIB LP library;these libraries contain many ill-conditioned and real-life problems.
Many machine learning tasks (e.g. metric and manifold learning problems) can be formulated as convex semidefinite programs. To enable the application of these tasks on a large-scale, scalability and computational effi...
详细信息
Many machine learning tasks (e.g. metric and manifold learning problems) can be formulated as convex semidefinite programs. To enable the application of these tasks on a large-scale, scalability and computational efficiency are considered as desirable properties for a practical semidefinite programming algorithm. In this paper, we theoretically analyze a new bilateral greedy optimization (denoted BILGO) strategy in solving general semidefinite programs on large-scale datasets. As compared to existing methods, BILGO employs a bilateral search strategy during each optimization iteration. In such an iteration, the current semidefinite matrix solution is updated as a bilateral linear combination of the previous solution and a suitable rank-1 matrix, which can be efficiently computed from the leading eigenvector of the descent direction at this iteration. By optimizing for the coefficients of the bilateral combination, BILGO reduces the cost function in every iteration until the KKT conditions are fully satisfied, thus, it tends to converge to a global optimum. In fact, we prove that BILGO converges to the global optimal solution at a rate of O(1/k), where k is the iteration counter. The algorithm thus successfully combines the efficiency of conventional rank-1 update algorithms and the effectiveness of gradient descent. Moreover, BILGO can be easily extended to handle low rank constraints. To validate the effectiveness and efficiency of BILGO, we apply it to two important machine learning tasks, namely Mahalanobis metric learning and maximum variance unfolding. Extensive experimental results clearly demonstrate that BILGO can solve large-scale semidefinite programs efficiently. (C) 2013 Elsevier B.V. All rights reserved.
Many chemical engineering processes involve a population of particles with a distribution of sizes that changes over time. Because calculating the time evolution of the full particle size distribution (PSD) is computa...
详细信息
Many chemical engineering processes involve a population of particles with a distribution of sizes that changes over time. Because calculating the time evolution of the full particle size distribution (PSD) is computationally expensive, it is common to instead calculate the time evolution of only finitely many moments of the distribution. The problem with moments is that they provide only a summary description of the PSD. In particular, they do not contain enough information to answer industrially relevant questions such as: How many particles are there in the size range [a, b]? What is the shape of the distribution? What is its D10? While these questions cannot be answered exactly, in this paper, we demonstrate that one can efficiently calculate rigorous bounds' on the answers by solving semidefinite programs. To the best of the authors' knowledge this natural application of semidefinite programming to PSDs has, until now, gone unnoticed. (C) 2017 Elsevier Ltd. All rights reserved.
This paper presents a placement algorithm for fault location observability using phasor measurement units (PMUs) in the presence or absence of zero injection buses. The problem is formulated as a binary semidefinite p...
详细信息
This paper presents a placement algorithm for fault location observability using phasor measurement units (PMUs) in the presence or absence of zero injection buses. The problem is formulated as a binary semidefinite programming (BSDP) model with binary decision variables, minimizing a linear objective function subject to linear matrix inequality (LMI) observability constraints. The model is extended to take into account the unavailability or limited capacity of communication links at some PMU installation buses. The BSDP problem is solved using an outer approximation scheme based on binary integer linear programming. The method is illustrated with a 6-bus test system. Numerical simulations are conducted on the IEEE 14-, 30-, and 57-bus standard test systems to verify the effectiveness of the proposed method.
This paper is devoted to the study of embedding methods for semidefinite programming problems using the duals formulated by Ramana, Tuncel, and Wolkowicz in 1997. Specifically, if we solve a semidefinite programming p...
详细信息
This paper is devoted to the study of embedding methods for semidefinite programming problems using the duals formulated by Ramana, Tuncel, and Wolkowicz in 1997. Specifically, if we solve a semidefinite programming problem (PD) (in either standard primal or dual form), a dual problem of (PD), which guarantees strong duality (i.e. a zero duality gap and dual attainment), is formulated. The semidefinite program (PD) and its newly formulated dual problem are then embedded in a larger problem. The embedding problem and its Lagrangian dual satisfy the generalized Slater conditions, and therefore, any path-following primal-dual interior-point method can be applied to solve this embedding problem pair. Like embedding methods appearing in the literature, a solution of the embedding problem can be used to extract the information about the original program (PD).
We introduce a new method of constructing approximation algorithms for combinatorial optimization problems using semidefinite programming. It consists of expressing each combinatorial object in the original problem as...
详细信息
We introduce a new method of constructing approximation algorithms for combinatorial optimization problems using semidefinite programming. It consists of expressing each combinatorial object in the original problem as a constellation of vectors in the semidefinite program. When we apply this technique to systems of linear equations mod p with at most two variables in each equation, we can show that the problem is approximable within (1 - kappa (p))p, where kappa (p)> 0 for all p. Using standard techniques we also show that it is NP-hard to approximate the problem within a constant ratio, independent of p. (C) 2001 Academic Press.
We consider a new semidefinite programming (SDP) relaxation of the symmetric traveling salesman problem (TSP) that may be obtained via an SDP relaxation of the more general quadratic assignment problem (QAP). We show ...
详细信息
We consider a new semidefinite programming (SDP) relaxation of the symmetric traveling salesman problem (TSP) that may be obtained via an SDP relaxation of the more general quadratic assignment problem (QAP). We show that the new relaxation dominates the one in [D. Cvetkovic, M. Cangalovic, and V. Kovacevic-Vujcic, semidefinite programming methods for the symmetric traveling salesman problem, in Proceedings of the 7th International IPCO Conference on Integer programming and Combinatorial Optimization, Springer-Verlag, London, UK, 1999, pp. 126-136]. Unlike the bound of Cvetkovic et al., the new SDP bound is not dominated by the Held-Karp linear programming bound, or vice versa.
When using interior methods for solving semidefinite programming (SDP), one needs to solve a system of linear equations at each iteration. For problems of large size, solving the system of linear equations can be very...
详细信息
When using interior methods for solving semidefinite programming (SDP), one needs to solve a system of linear equations at each iteration. For problems of large size, solving the system of linear equations can be very expensive. In this paper, based on a semismooth equation reformulation using Fischer's function, we propose a filter method with trust region for solving large-scale SDP problems. At each iteration we perform a number of conjugate gradient iterations, but do not need to solve a system of linear equations. Under mild assumptions, the convergence of this algorithm is established. Numerical examples are given to illustrate the convergence results obtained.
暂无评论