Solving Reduced KKT Systems in Barrier Methods for Linear and Quadratic Programming

Gill, Philip E.; Murray, Walter; Ponceleón, Dulce B.; Saunders, Michael A.

doi:10.21236/ada239191

Cited by 55 publications

(42 citation statements)

References 31 publications

(42 reference statements)

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“…(For a discussion of the role of regularization in linear and quadratic programming, see, e.g., [34,35]). If the second-order sufficient conditions hold and the gradients of the active constraints are linearly independent, then for µ sufficiently small, a differentiable primal-dual trajectory of solutions x(µ), y(µ) exists and converges to x * , y * as µ → 0 + .…”

Section: Primal-dual Methodsmentioning

confidence: 99%

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A primal-dual trust region algorithm for nonlinear optimization

Gertz

Gill

2004

Math. Program., Ser. B

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Abstract. This paper concerns general (nonconvex) nonlinear optimization when first and second derivatives of the objective and constraint functions are available. The proposed method is based on finding an approximate solution of a sequence of unconstrained subproblems parameterized by a scalar parameter. The objective function of each unconstrained subproblem is an augmented penalty-barrier function that involves both primal and dual variables. Each subproblem is solved using a second-derivative Newton-type method that employs a combined trust region and line search strategy to ensure global convergence. It is shown that the trustregion step can be computed by factorizing a sequence of systems with diagonally-modified primal-dual structure, where the inertia of these systems can be determined without recourse to a special factorization method. This has the benefit that off-the-shelf linear system software can be used at all times, allowing the straightforward extension to large-scale problems. Numerical results are given for problems in the COPS test collection.

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Section: Primal-dual Methodsmentioning

confidence: 99%

“…The efficiency of a direct sparse solver should be less dependent on H + J T W −1 J being sparse (see, e.g., [29,34]). In this situation it is customary to symmetrize the system so that a symmetric solver can be used.…”

Section: Definition Of the Inner Iterationsmentioning

confidence: 99%

A primal-dual trust region algorithm for nonlinear optimization

Gertz

Gill

2004

Math. Program., Ser. B

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“…In this case, the sequence would be a strictly positive approximation to x*. [GMPS91] (see also Lustig [Lus88]). Methods based on the solution of (7.2) are usually referred to as primal-dual algorithms because both x and z are maintained to be positive.…”

Section: Summary Of Primal Methodsmentioning

confidence: 99%

“…All the primal-dual algorithms have very similar theoretical properties, but only the primal-dual algorithm of Section 7.1 has been used in the principal known implementations [LMS89,LMS90,Meh9O,GMPS91]. The key system of equations is "less nonlinear" than for the other three variations.…”

Section: Further Commentsmentioning

confidence: 99%

Primal—dual methods for linear programming

Gill

Murray

Ponceleón

et al. 1995

Mathematical Programming

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Many interior-point methods for linear programming are based on the properties of the logarithmic barrier function. We first give a convergence proof for the (primal) projected Newton barrier method. We then analyze three types of barrier method that can be categorized as primal, dual and primal-dual. All three approaches may be derived from the application of Newton's method to different variants of the same system of nonlinear equations. A fourth variant of the same equations leads to a new primal-dual algorithm.In each of the methods discussed, convergence is demonstrated without the need for a nondegeneracy assumption. In particular, convergence is established for a primal-dual algorithm that allows a different step in the primal and dual variables.Finally, we describe a new method for treating free variables.

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“…In our paper, we reformulate the linear program to ensure the constraint matrix is very sparse so that the problem can be solved efficiently. The computational advantage of using sparse constraint matrices has been well established in the linear programming literature [25,31,39,42]. While most optimization research in compressed sensing focuses on creating customized algorithms, our approach uses existing algorithms but a sparser problem formulation to speed up computation.…”

Section: Introduction and Contribution Overviewmentioning

confidence: 99%

Revisiting compressed sensing: exploiting the efficiency of simplex and sparsification methods

et al. 2016

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We propose two approaches to solve large-scale compressed sensing problems. The first approach uses the parametric simplex method to recover very sparse signals by taking a small number of simplex pivots, while the second approach reformulates the problem using Kronecker products to achieve faster computation via a sparser problem formulation. In particular, we focus on the computational aspects of these methods in compressed sensing. For the first approach, if the true signal is very sparse and we initialize our solution to be the zero vector, then a customized parametric simplex method usually takes a small number of iterations to converge. Our numerical studies show that this approach is 10 times faster than state-of-the-art methods for recovering very sparse signals. The second approach can be used when the sensing matrix is the Kronecker product of two smaller matrices. We show that the best-known sufficient condition for the Kronecker compressed sensing (KCS) strategy to obtain a perfect recovery is more restrictive than the corresponding condition if using the first approach. However, KCS can be formulated as a linear program with a very sparse constraint matrix, whereas the first approach involves a completely dense constraint matrix. Hence, algorithms that benefit from sparse problem representation, such as interior point methods (IPMs), are expected to have computational advantages for the KCS problem. We numerically demonstrate that KCS combined with IPMs is up to 10The first author's research is supported by ONR Award N00014-13-1-0093, the third author's by NSF Grant III-1116730, and the fourth author's by NSF Grant DMS-1005539.

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Solving Reduced KKT Systems in Barrier Methods for Linear and Quadratic Programming

Cited by 55 publications

References 31 publications

A primal-dual trust region algorithm for nonlinear optimization

A primal-dual trust region algorithm for nonlinear optimization

Primal—dual methods for linear programming

Revisiting compressed sensing: exploiting the efficiency of simplex and sparsification methods

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