Semidefinite programming

Helmberg, Christoph

doi:10.1016/s0377-2217(01)00143-6

Cited by 59 publications

(41 citation statements)

References 39 publications

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“…One immediate consequence of the SDP formulation (3) is that we can numerically solve the FMMP problem efficiently, using standard algorithms for an SDP (e.g., [23,2,1,11,24]; see [13] for a comprehensive list of current SDP software). For m no more than 1000 or so, interior-point methods can be used to solve the FMMP problem in a minute or so (or much less) on a small personal computer.…”

Section: Semidefinite Program (Sdp) Formulationmentioning

confidence: 99%

The Fastest Mixing Markov Process on a Graph and a Connection to a Maximum Variance Unfolding Problem

et al. 2006

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Abstract. We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue λ 2 of the Laplacian of the weighted graph. In this paper we consider the problem of assigning transition rates to the edges so as to maximize λ 2 subject to a linear constraint on the rates. This is the problem of finding the fastest mixing Markov process (FMMP) on the graph. We show that the FMMP problem is a convex optimization problem, which can in turn be expressed as a semidefinite program, and therefore effectively solved numerically. We formulate a dual of the FMMP problem and show that it has a natural geometric interpretation as a maximum variance unfolding (MVU) problem, i.e., the problem of choosing a set of points to be as far apart as possible, measured by their variance, while respecting local distance constraints. This MVU problem is closely related to a problem recently proposed by Weinberger and Saul as a method for "unfolding" high-dimensional data that lies on a low-dimensional manifold. The duality between the FMMP and MVU problems sheds light on both problems, and allows us to characterize and, in some cases, find optimal solutions.

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Section: Semidefinite Program (Sdp) Formulationmentioning

confidence: 99%

The Fastest Mixing Markov Process on a Graph and a Connection to a Maximum Variance Unfolding Problem

et al. 2006

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“…We refer the reader to the survey provided in [16] for a more comprehensive introduction to the field of semidefinite programming. We denote S n the set of symmetric matrices of size n. The standard scalar product between two matrices A and B in S n is defined as:…”

Section: Notation and Definitionsmentioning

confidence: 99%

“…As explained in [16], semidefinite programming is linear programming over the cone of positive semidefinite matrices S + n . A semidefinite program thus involves a matrix variable X ∈ S + n and deals with the maximization of a linear function of X subject to a series of constraints whose expression are also linear with respect to X. Symmetric matrices C, A 1 , ...A M are used to formulate the objective function and the technical constraints.…”

Section: Notation and Definitionsmentioning

confidence: 99%

An evaluation of semidefinite programming based approaches for discrete lot-sizing problems

Gicquel

Lisser

Minoux

2014

European Journal of Operational Research

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The present work is intended as a first step towards applying semidefinite programming models and tools to discrete lot-sizing problems including sequence-dependent changeover costs and times. Such problems can be formulated as quadratically constrained quadratic binary programs. We investigate several semidefinite relaxations by combining known reformulation techniques recently proposed for generic quadratic binary problems with problemspecific strengthening procedures developped for lot-sizing problems. Our computational results show that the semidefinite relaxations consistently provide lower bounds of significantly improved quality as compared with those provided by the best previously published linear relaxations. In particular, the gap between the semidefinite relaxation and the optimal integer solution value can be closed for a significant proportion of the small-size instances, thus avoiding to resort to a tree search procedure. The reported computation times are significant. However improvements in SDP technology can still be expected in the future, making SDP based approaches to discrete lot-sizing

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“…Therefore, we must graduate to a richer language, that of semidefinite programming. Every semidefinite program with an N × N constraint matrix can be solved in time O(N 3 ) [18]. A semidefinite program is any problem of the form…”

Section: Sdp Formulationmentioning

confidence: 99%

Assessing the vulnerability of replicated network services

Bissias

Levine

Sitaraman

2010

Proceedings of the 6th International COnference

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Client-server networks are pervasive, fundamental, and include such key networks as the Internet, power grids, and road networks. In a client-server network, clients obtain a service by connecting to one of a redundant set of servers. These networks are vulnerable to node and link failures, causing some clients to become disconnected from the servers. We develop algorithms that quantify and bound the inherent vulnerability of a clientserver network using semidefinite programming (SDP) and branch-and-cut techniques. Further, we develop a divide-and-conquer algorithm that solves the problem for large graphs. We use these techniques to show that: for the Philippine Power Grid removing just over 6% of the transmission lines will disconnect at least 20% but not more than 50% of the substations from all generators; on a large wireless mesh network disrupting 5% of wireless links between relays removes Internet access for half the relays; even after any 16% of Tier 2 ASes are removed, more than 50% of the remaining Tier 2 ASes will be connected to the Tier 1 backbone; when 300 roadblocks are erected in Michigan, it's possible to disconnect 28-43% of the population from all airports.

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Semidefinite programming

Cited by 59 publications

References 39 publications

The Fastest Mixing Markov Process on a Graph and a Connection to a Maximum Variance Unfolding Problem

The Fastest Mixing Markov Process on a Graph and a Connection to a Maximum Variance Unfolding Problem

An evaluation of semidefinite programming based approaches for discrete lot-sizing problems

Assessing the vulnerability of replicated network services

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