Multilevel Algorithms for Large-Scale Interior Point Methods

Benzi, Michele; Haber, Eldad; Taralli, Lauren

doi:10.1137/060650799

Cited by 12 publications

(21 citation statements)

References 30 publications

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“…With a proper block partitioning, K 3 can be cast into a KKT formulation as well. Under our assumptions on H and J , and as long as X and Z are diagonal with positive entries, K 2 and K 3 are nonsingular; K 2 has n positive and m negative eigenvalues, while K 3 has n positive and n + m negative eigenvalues, see, e.g., , Lemma 4.1], , Lemma 3.5, 3.8]. The distinctive feature of these two matrices is their behavior in the limit of the IP procedure.…”

Section: Preliminariesmentioning

confidence: 96%

“…Interior Point (IP) methods are effective iterative procedures for solving linear, quadratic, and nonlinear programming problems, possibly of very large dimension, see and references therein. Because they are second‐order methods, a linear algebra phase constitutes their computational core, and its practical implementation is crucial for the efficiency of the overall optimization procedure.…”

Section: Introductionmentioning

confidence: 99%

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Spectral estimates for unreduced symmetric KKT systems arising from Interior Point methods

Morini

Simoncini

Tani

2016

Numerical Linear Algebra App

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We consider symmetrized Karush-Kuhn-Tucker systems arising in the solution of convex quadratic programming problems in standard form by Interior Point methods. Their coefficient matrices usually have 3 3 block structure, and under suitable conditions on both the quadratic programming problem and the solution, they are nonsingular in the limit. We present new spectral estimates for these matrices: the new bounds are established for the unpreconditioned matrices and for the matrices preconditioned by symmetric positive definite augmented preconditioners. Some of the obtained results complete the analysis recently given by ‡ In the context of IP methods, the coefficient matrix of the system to be solved at each iteration is often referred to as the barrier KKT matrix. Following [7, p. 92] we omit the term "barrier" for simplicity and denote the system as KKT system, see also [8,9].B. MORINI, V. SIMONCINI AND M. TANIwhere J 2 R m n has full row rank m < n, H 2 R n n is symmetric and positive semidefinite (SPSD in the following), x;´; c 2 R n , y; b 2 R m , and the inequalities are meant componentwise. The application of a primal-dual IP method gives rise, at each iteration, to an unsymmetric 3 3 block matrix of dimension 2n C m, which allows for alternative formulations of differing dimension, conditioning and definiteness [3,[9][10][11]. In fact, the unsymmetric 3 3 matrix can be easily symmetrized without increasing the conditioning of the system [7], and here, we will refer to the resulting symmetric matrix as the unreduced KKT matrix. On the other hand, by exploiting the structure of the unsymmetric 3 3 matrix and block elimination, it is common to use a linear system of dimension n C m with a reduced (or augmented) symmetric 2 2 KKT matrix. Finally, a further block elimination may yield a condensed system (or normal equations) where the matrix is a Schur complement of dimension n. The focus of this work is the theoretical study of the unreduced 3 3 formulation and the numerical illustration of the obtained results. Unlike the reduced 2 2 matrix, under suitable conditions on both the problem (1) and the solution, the unreduced KKT matrix has condition number asymptotically uniformly bounded and typically remains well-conditioned as the solution is approached [3,7]. Motivated by this feature, in a very recent paper, Greif et al. [12] presented a spectral analysis for the 3 3 matrix and claimed that this formulation can be preferable to the reduced one in terms of eigenvalues and conditioning, although a 'benign' asymptotically ill-conditioned scaling has to be applied to the right-hand side and to the system variables [3]. Such a study covers also variants of KKT matrices arising from regularization of the optimization problem.The study conducted in [12] has renewed the interest in the unreduced formulation but leaves some issues open that are worth investigating and are a prerequisite for a thorough comparison of the 2 2 and 3 3 formulations. Specifically, some eigenvalue bounds presented in [12] may be overly pe...

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Section: Preliminariesmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Spectral estimates for unreduced symmetric KKT systems arising from Interior Point methods

Morini

Simoncini

Tani

2016

Numerical Linear Algebra App

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“…Naturally these objective function and constraints involve many parameters and the experts may assign them different values ( [1,2,3,4]). After the publication of the Dantzig and Wolfe decomposition method [7], there have been numerous subsequent works on large scale linear and nonlinear programming problems with block angular structure ( [1,6,12]). Abo Sinna et al [4] extended the technique for order preference by similarity ideal solution (TOPSIS) to solve multi-objective large scale non-linear programming problem.…”

Section: Introductionmentioning

confidence: 99%

“…El-Sawy et al [9] introduced an algorithm for decomposing the parametric space in large scale linear vector optimization problems under fuzzy environment. Recently, notable studies have been done in the area of multi-level and multi-objective large scale programming problems ( [3,4,6,12]). Benzi et al [6] developed and compared multi-level algorithms for solving large scale bound constrained nonlinear problems via interior point methods.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, notable studies have been done in the area of multi-level and multi-objective large scale programming problems ( [3,4,6,12]). Benzi et al [6] developed and compared multi-level algorithms for solving large scale bound constrained nonlinear problems via interior point methods. It show how a multilevel continuation strategy can be used to obtain good initial guesses for each nonlinear iteration.…”

Section: Introductionmentioning

confidence: 99%

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A Decomposition Algorithm for Solving a Three Level Large Scale linear Programming Problem

Sultan¹,

Emam²,

Abohany³

2014

Appl. Math. Inf. Sci.

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This paper presents a three level large scale linear programming problem in which the objective functions at every level are to be maximized. A three level programming problem can be thought as a static version of the Stackelberg strategy. An algorithm for solving a three planner model and a solution method for treating this problem are suggested. At each level we attempt to optimize its problem separately as a large scale programming problem using Dantzig and Wolfe decomposition method. Therefore, we handle the optimization process through a series of sub problems that can be solved independently. Finally, a numerical example is given to clarify the main results developed in this paper.

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