An ε-sensitivity analysis in the primal–dual interior point method

Kim, Woo-Je; Park, Chan-Kyoo; Park, Soondal

doi:10.1016/s0377-2217(98)00117-9

Cited by 10 publications

(7 citation statements)

References 6 publications

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“…Similar results, but changing only one of the primal or dual near-optimal solutions, were obtained by Kim, Park and Park [8].…”

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confidence: 78%

An Interior-Point Approach to Sensitivity Analysis in Degenerate Linear Programs

Yıldırım¹,

Todd²

2002

SIAM J. Optim.

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Abstract. We consider the interior-point approach to sensitivity analysis in linear programming developed by the authors. We investigate the quality of the interior-point bounds under degeneracy. In the case of a special degeneracy, we show that these bounds have the same nice asymptotic relationship with the optimal partition bounds as in the nondegenerate case. We prove a weaker relationship for general degenerate linear programs.Key words. sensitivity analysis, degeneracy, interior-point methods, linear programming AMS subject classifications. 90C31,90C51,90C051. Introduction. Sensitivity analysis (or post-optimality analysis) is the study of how the optimal solution of an optimization problem changes with respect to the changes in the problem data. The possible presence of errors in the problem data often makes sensitivity analysis as important as solving the original problem itself.In the context of linear programming, sensitivity analysis can be performed using an optimal basis approach (as in the simplex method) or an optimal partition approach, where the optimal partition refers to knowing, for each index, whether the corresponding component of an optimal primal solution or of an optimal dual slack vector can be positive. The latter approach has close connections with interior-point methods since such methods, when properly terminated, provide an optimal solution in the relative interior of the optimal face, from which the optimal partition is readily available. In fact, as will shortly be discussed in more detail, the optimal partition approach has been developed by Adler and Monteiro [1] and Jansen, de Jong, Roos and Terlaky [7] as a promising alternative in order to circumvent the drawbacks of the classical optimal basis approach in the presence of degeneracy. Later, Monteiro and Mehrotra [9] extended this approach by relaxing the requirement that the optimal partition be known. They also provided two methods to estimate the range of perturbations, each of which can be performed at any optimal solution, regardless of where it lies on the optimal face. More recently, Greenberg, Holder, Roos and Terlaky [5] related the dimension of the optimal set to the dimension of the set of objective perturbations for which the optimal partition is invariant. Greenberg [4] considered the simultaneous perturbations of the right-hand side and the cost vectors from an optimal partition perspective.In [13], the authors studied perturbations of the right-hand side and the cost parameters in linear programming, motivated by how interior-point methods from a near-optimal pair of strictly feasible solutions for a problem and its dual would compare with the optimal basis approach obtained from a nondegenerate optimal basic solution for such perturbations. The proposed interior-point perspective stems from the objectives of regaining feasibility and maintaining near-optimality in a single it-

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“…Similar results, but changing only one of the primal or dual near-optimal solutions, were obtained by Kim, Park and Park [8].…”

supporting

confidence: 78%

An Interior-Point Approach to Sensitivity Analysis in Degenerate Linear Programs

Yıldırım¹,

Todd²

2002

SIAM J. Optim.

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“…After this paper was written (and revised), we became aware of a related paper by Kim, Park, and Park [13], henceforth KPP. The authors also consider changes in the right-hand side or the cost parameters and investigate when a single interior-point-like step from a nearoptimal pair of strictly feasible solutions for a problem and its dual can regain feasibility and maintain near-optimality.…”

Section: Sensitivity Analysis In Lp and Sdpmentioning

confidence: 99%

Sensitivity analysis in linear programming and semidefinite programming using interior-point methods

Yıldırım

Todd

2001

Math. Program.

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Abstract. We analyze perturbations of the right-hand side and the cost parameters in linear programming (LP) and semidefinite programming (SDP). We obtain tight bounds on the perturbations that allow interior-point methods to recover feasible and near-optimal solutions in a single interior-point iteration. For the unique, nondegenerate solution case in LP, we show that the bounds obtained using interior-point methods compare nicely with the bounds arising from using the optimal basis. We also present explicit bounds for SDP using the MonteiroZhang family of search directions and specialize them to the AHO, H..K..M, and NT directions.

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“…After this publication, many researchers have published on similar subjects exploring different kinds of data perturbations within the context of LP (see for instance [13,17,18,27,35,36,46,56,57]). With the advent of interior point methods (IPM), other researchers have extended sensitivity analysis to explore how central-path solutions and Newton iterates behave under data perturbations for LP and semidefinite programming (SDP) (see among others [22,62,63]). Another aspect of the perturbation theory that many authors have studied is the analytic properties of the optimal solutions and the optimal objective values of the perturbed data instances such as the continuity, the differentiability, the Lipschitz conditions and bounds, etc.…”

Section: Applicationsmentioning

confidence: 99%

Extreme points of well-posed polytopes

Núñez

2006

Optimization

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Given a real matrix, we study the problem of finding a minimal set of columns spanning the convex polytope generated by the columns of the matrix. By considering the matrix as a data instance subject to perturbations, we introduce the notion of ill-posedness of a polytope, in the sense that small perturbations of the matrix might yield matrices with different combinatorial structures. We relate this notion of the ill-posedness to the better-known notion of the ill-posedness of conic homogeneous systems and propose a characterization of the ill-posed matrices. We introduce a new condition measure for a matrix and study the complexity of a solution algorithm in terms of this condition measure. The algorithm has a polynomial bound on the number of iterations to identify a set of columns corresponding to the extreme points of a polytope. The complexity bound depends on the condition measure and the size of the matrix.

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An ε-sensitivity analysis in the primal–dual interior point method

Cited by 10 publications

References 6 publications

An Interior-Point Approach to Sensitivity Analysis in Degenerate Linear Programs

An Interior-Point Approach to Sensitivity Analysis in Degenerate Linear Programs

Sensitivity analysis in linear programming and semidefinite programming using interior-point methods

Extreme points of well-posed polytopes

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