Cones of diagonally dominant matrices

Barker, George Phillip; Carlson, David

doi:10.2140/pjm.1975.57.15

Cited by 107 publications

(68 citation statements)

References 7 publications

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“…In this inner product space, the set S of symmetric matrices form a closed subspace and SDD + is a closed and convex polyhedral cone [1,18,16]. Therefore, the feasible region is a closed and convex set in IR ncols×ncols .…”

Section: Test Problemmentioning

confidence: 99%

“…Throughout this paper we assume that f is defined and has continuous partial derivatives on an open set that contains Ω. The Spectral Projected Gradient (SPG) method [6,7] was recently proposed for solving (1), especially for large-scale problems since the storage requirements are minimal. This method has proved to be effective for very large-scale convex programming problems.…”

Section: Introductionmentioning

confidence: 99%

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Inexact spectral projected gradient methods on convex sets

Birgin¹

2003

IMA Journal of Numerical Analysis

162

178

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A new method is introduced for large scale convex constrained optimization. The general model algorithm involves, at each iteration, the approximate minimization of a convex quadratic on the feasible set of the original problem and global convergence is obtained by means of nonmonotone line searches. A specific algorithm, the Inexact Spectral Projected Gradient method (ISPG), is implemented using inexact projections computed by Dykstra's alternating projection method and generates interior iterates. The ISPG method is a generalization of the Spectral Projected Gradient method (SPG), but can be used when projections are difficult to compute. Numerical results for constrained least-squares rectangular matrix problems are presented.

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Section: Test Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Inexact spectral projected gradient methods on convex sets

Birgin¹

2003

IMA Journal of Numerical Analysis

162

178

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“…Since L need not be all of R m , the matrix D may be singular in which case Slater's condition fails for (P). Let We need only show that, see Barker and Carlson (1975) which proves (7.7). In fact, we get that (7.11) ~g(F)=S'.…”

Section: Applications In Matrix Theorymentioning

confidence: 84%

Characterization of optimality for the abstract convex program with finite dimensional range

Borwein

1981

J. Aust. Math. Soc.

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This paper presents characterizations of optimality for the abstract convex program(E-S,x when S is an arbitrary convex cone in a finite dimensional space, U is a convex set and/; and g are respectively convex and S-convex (on Q). These characterizations, which include a Lagrange multiplier theorem and do not presume any a priori constraint qualification, subsume those presently in the literature.1980 Mathematics subject classification (Amer. Math. Soc.): primary 90 C 25.

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“…Since all sos polynomials are nonnegative, the optimal value of the SDP in (9) is a lower bound to the optimal value of the optimization problem in (8). Unfortunately, before solving the SDP, we do not have access to the vectors u i in the decomposition of the optimal matrix Y .…”

Section: Nonconvex Polynomial Optimizationmentioning

confidence: 99%

Optimization over structured subsets of positive semidefinite matrices via column generation

Ahmadi

Dash²,

Hall

2017

Discrete Optimization

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We develop algorithms to construct inner approximations of the cone of positive semidefinite matrices via linear programming and second order cone programming. Starting with an initial linear algebraic approximation suggested recently by Ahmadi and Majumdar, we describe an iterative process through which our approximation is improved at every step. This is done using ideas from column generation in large-scale linear programming. We then apply these techniques to approximate the sum of squares cone in a nonconvex polynomial optimization setting, and the copositive cone for a discrete optimization problem.

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Cones of diagonally dominant matrices

Cited by 107 publications

References 7 publications

Inexact spectral projected gradient methods on convex sets

Inexact spectral projected gradient methods on convex sets

Characterization of optimality for the abstract convex program with finite dimensional range

Optimization over structured subsets of positive semidefinite matrices via column generation

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