Necessary and sufficient global optimality conditions for NLP reformulations of linear SDP problems

Grippo, Luigi; Palagi, Laura; Piccialli, Veronica

doi:10.1007/s10898-008-9328-4

Cited by 7 publications

(12 citation statements)

References 11 publications

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“…In what follows, we recall the main results by Burer and Monteiro [14] and Grippo et al [16]. These results give necessary and sufficient conditions that a (local) solution to the QCQP arising from the change of variables K = YY T with Y ∈ R n×d such that d ≤ n yields a solution to the standard form SDP…”

Section: Optimality Conditionsmentioning

confidence: 85%

“…A key assumption made in [16] is that the SDP (5) and its dual have nonempty solution sets and the gap between the primal and dual solutions of (5) is zero. The dual of problem (5) is…”

Section: Optimality Conditionsmentioning

confidence: 99%

“…for any matrices A ∈ S n , B ∈ R n×m and the vector b = vec(B [16] give the following sufficient condition for a solution of (8) to be a solution of the SDP (5) and also a global solution to (8). For the KKT optimality conditions, we refer to [5].…”

Section: Optimality Conditionsmentioning

confidence: 99%

“…Since problem (15) is obtained from problem (16) by the variable substitution K = YY T that yields a positive semidefinite matrix, any feasible matrix Y for problem (15) with any dimension d clearly yields a feasible matrix K = YY T for problem (16). Thus, any solution algorithm applied to problem (2) or (15) or an equivalent formulation is guaranteed to give a bounded solution whenever Assumption 3.2 is satisfied.…”

Section: Strong Duality Of the Graph Embedding Sdpmentioning

confidence: 99%

“…Based on the theory of SDPs and their low-rank quadratic formulations developed in [13,14,16,21], we show that problem (2) is in fact a low-rank formulation of problem (3). Motivated by this fact, we develop an incremental low-rank method for solving the SDP (3).…”

Section: Introductionmentioning

confidence: 99%

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Finding graph embeddings by incremental low-rank semidefinite programming

Pulkkinen

2015

Optimization Methods and Software

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Finding a low-dimensional embedding of a graph of n nodes in R d is an essential task in many applications. For instance, maximum variance unfolding (MVU) is a well-known dimensionality reduction method that involves solving this problem. The standard approach is to formulate the embedding problem as a semidefinite program (SDP). However, the SDP approach does not scale well to large graphs. In this paper, we exploit the fact that many graphs have an intrinsically low dimension, and thus the optimal matrix resulting from the solution of the SDP has a low rank. This observation leads to a quadratic reformulation of the SDP that has far fewer variables, but on the other hand, is a difficult convex maximization problem. We propose an approach for obtaining a solution to the SDP by solving a sequence of smaller quadratic problems with increasing dimension. Utilizing an augmented Lagrangian and an interior-point method for solving the quadratic problems, we demonstrate with numerical experiments on MVU problems that the proposed approach scales well to very large graphs.

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Section: Optimality Conditionsmentioning

confidence: 85%

“…A key assumption made in [16] is that the SDP (5) and its dual have nonempty solution sets and the gap between the primal and dual solutions of (5) is zero. The dual of problem (5) is…”

Section: Optimality Conditionsmentioning

confidence: 99%

Section: Optimality Conditionsmentioning

confidence: 99%

Section: Strong Duality Of the Graph Embedding Sdpmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Finding graph embeddings by incremental low-rank semidefinite programming

Pulkkinen

2015

Optimization Methods and Software

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Computational Approaches to Max-Cut

Palagi

Piccialli

Rendl

et al. 2011

International Series in Operations Research &Amp; Management Science

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An unconstrained minimization method for solving low-rank SDP relaxations of the maxcut problem

2009

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In this paper we consider low-rank semidefinite programming (LRSDP) relaxations of combinatorial quadratic problems that are equivalent to the maxcut problem. Using the Gramian representation of a positive semidefinite matrix, the LRSDP problem can be formulated as the nonconvex nonlinear programming problem of minimizing a quadratic function with quadratic equality constraints. For the solution of this problem we propose a continuously differentiable exact merit function that exploits the special structure of the constraints and we use this function to define an efficient and globally convergent algorithm. Finally, we test our code on an extended set of instances of the maxcut problem and we report comparisons with other existing codes. © 2009 Springer and Mathematical Programming Society

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Necessary and sufficient global optimality conditions for NLP reformulations of linear SDP problems

Cited by 7 publications

References 11 publications

Finding graph embeddings by incremental low-rank semidefinite programming

Finding graph embeddings by incremental low-rank semidefinite programming

Computational Approaches to Max-Cut

An unconstrained minimization method for solving low-rank SDP relaxations of the maxcut problem

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