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53
Interiorpoint Methods
, 2000
"... The modern era of interiorpoint methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadrati ..."
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Cited by 612 (15 self)
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The modern era of interiorpoint methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semidefinite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. We review some of the key developments in the area, including comments on both the complexity theory and practical algorithms for linear programming, semidefinite programming, monotone linear complementarity, and convex programming over sets that can be characterized by selfconcordant barrier functions.
LOQO: An interior point code for quadratic programming
, 1994
"... ABSTRACT. This paper describes a software package, called LOQO, which implements a primaldual interiorpoint method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex ..."
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Cited by 194 (10 self)
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ABSTRACT. This paper describes a software package, called LOQO, which implements a primaldual interiorpoint method for general nonlinear programming. We focus in this paper mainly on the algorithm as it applies to linear and quadratic programming with only brief mention of the extensions to convex and general nonlinear programming, since a detailed paper describing these extensions were published recently elsewhere. In particular, we emphasize the importance of establishing and maintaining symmetric quasidefiniteness of the reduced KKT system. We show that the industry standard MPS format can be nicely formulated in such a way to provide quasidefiniteness. Computational results are included for a variety of linear and quadratic programming problems. 1.
NonInterior Continuation Methods For Solving Semidefinite Complementarity Problems
 Math. Programming
, 1999
"... There recently has been much interest in noninterior continuation/smoothing methods for solving linear/nonlinear complementarity problems. We describe extensions of such methods to complementarity problems defined over the cone of blockdiagonal symmetric positive semidefinite real matrices. These ..."
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Cited by 43 (6 self)
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There recently has been much interest in noninterior continuation/smoothing methods for solving linear/nonlinear complementarity problems. We describe extensions of such methods to complementarity problems defined over the cone of blockdiagonal symmetric positive semidefinite real matrices. These extensions involve the ChenMangasarian class of smoothing functions and the smoothed FischerBurmeister function. Issues such as existence of Newton directions, boundedness of iterates, global convergence, and local superlinear convergence will be studied. Preliminary numerical experience on semidefinite linear programs is also reported. Key words. Semidefinite complementarity problem, smoothing function, noninterior continuation, global convergence, local superlinear convergence. 1 Introduction There recently has been much interest in semidefinite linear programs (SDLP) and, more generally, semidefinite linear complementarity problems (SDLCP), which are extensions of LP and LCP, respecti...
A Study of Search Directions in PrimalDual InteriorPoint Methods for Semidefinite Programming
, 1998
"... We discuss several di#erent search directions which can be used in primaldual interiorpoint methods for semidefinite programming problems and investigate their theoretical properties, including scale invariance, primaldual symmetry, and whether they always generate welldefined directions. Among ..."
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Cited by 35 (1 self)
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We discuss several di#erent search directions which can be used in primaldual interiorpoint methods for semidefinite programming problems and investigate their theoretical properties, including scale invariance, primaldual symmetry, and whether they always generate welldefined directions. Among the directions satisfying all but at most two of these desirable properties are the AlizadehHaeberlyOverton, HelmbergRendl VanderbeiWolkowicz/KojimaShindohHara/Monteiro, NesterovTodd, Gu, and Toh directions, as well as directions we will call the MTW and Half directions. The first five of these appear to be the best in our limited computational testing also. Key words: semidefinite programming, search direction, invariance properties. AMS Subject classification: 90C05. Abbreviated title: Search directions in SDP 1 Introduction This paper is concerned with interiorpoint methods for semidefinite programming (SDP) problems and in particular the various search directions they use and ...
A Primaldual InteriorPoint Method for Linear Optimization Based on a New Proximity Function
, 2002
"... In this paper we present a generic primaldual interiorpoint algorithm for linear optimization in which the search direction depends on a univariate kernel function which is also used as proximity measure in the analysis of the algorithm. We present some powerful tools for the analysis of the alg ..."
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Cited by 35 (9 self)
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In this paper we present a generic primaldual interiorpoint algorithm for linear optimization in which the search direction depends on a univariate kernel function which is also used as proximity measure in the analysis of the algorithm. We present some powerful tools for the analysis of the algorithm under the assumption that the kernel function satisfies three easy to check and mild conditions (i.e., exponential convexity, superconvexity and monotonicity of the second derivative). The approach is demonstrated by introducing a new kernel function and showing that the corresponding largeupdate algorithm improves the iteration complexity with a factor n 1 4 when compared with the classical method, which is based on the use of the logarithmic barrier function.
Polynomial Convergence of a New Family of PrimalDual Algorithms for Semidefinite Programming
, 1996
"... This paper establishes the polynomial convergence of a new class of (feasible) primaldual interiorpoint path following algorithms for semidefinite programming (SDP) whose search directions are obtained by applying Newton method to the symmetric central path equation (P T XP ) 1=2 (P \Gamma1 ..."
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Cited by 32 (11 self)
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This paper establishes the polynomial convergence of a new class of (feasible) primaldual interiorpoint path following algorithms for semidefinite programming (SDP) whose search directions are obtained by applying Newton method to the symmetric central path equation (P T XP ) 1=2 (P \Gamma1 SP \GammaT )(P T XP ) 1=2 \Gamma I = 0; where P is a nonsingular matrix. Specifically, we show that the shortstep path following algorithm based on the Frobenius norm neighborhood and the semilongstep path following algorithm based on the operator 2norm neighborhood have O( p nL) and O(nL) iterationcomplexity bounds, respectively. When P = I, this yields the first polynomially convergent semilongstep algorithm based on a pure Newton direction. Restricting the scaling matrix P at each iteration to a certain subset of nonsingular matrices, we are able to establish an O(n 3=2 L) iterationcomplexity for the longstep path following method. The resulting subclass of search direct...
A fullNewton step O(n) infeasible interiorpoint algorithm for linear optimization
, 2005
"... We present a primaldual infeasible interiorpoint algorithm. As usual, the algorithm decreases the duality gap and the feasibility residuals at the same rate. Assuming that an optimal solution exists it is shown that at most O(n) iterations suffice to reduce the duality gap and the residuals by the ..."
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Cited by 19 (7 self)
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We present a primaldual infeasible interiorpoint algorithm. As usual, the algorithm decreases the duality gap and the feasibility residuals at the same rate. Assuming that an optimal solution exists it is shown that at most O(n) iterations suffice to reduce the duality gap and the residuals by the factor 1/e. This implies an O(nlog(n/ε)) iteration bound for getting an εsolution of the problem at hand, which coincides with the best known bound for infeasible interiorpoint algorithms. The algorithm constructs strictly feasible iterates for a sequence of perturbations of the given problem and its dual problem. A special feature of the algorithm is that it uses only fullNewton steps. Two types of fullNewton steps are used, socalled feasibility steps and usual (centering) steps. Starting at strictly feasible iterates of a perturbed pair, (very) close its central path, feasibility steps serve to generate strictly feasible iterates for the next perturbed pair. By accomplishing a few centering steps for the new perturbed pair we obtain strictly feasible iterates close enough to the central path of the new perturbed pair. The algorithm finds an optimal solution or detects infeasibility or unboundedness of the given problem.
Approximate Farkas Lemmas and Stopping Rules for Iterative InfeasiblePoint Algorithms for Linear Programming
, 1996
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Probabilistic Analysis of an InfeasibleInteriorPoint Algorithm for Linear Programming
, 1998
"... We consider an infeasibleinteriorpoint algorithm, endowed with a finite termination scheme, applied to random linear programs generated according to a model of Todd. Such problems have degenerate optimal solutions, and possess no feasible starting point. We use no information regarding an optimal ..."
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Cited by 14 (3 self)
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We consider an infeasibleinteriorpoint algorithm, endowed with a finite termination scheme, applied to random linear programs generated according to a model of Todd. Such problems have degenerate optimal solutions, and possess no feasible starting point. We use no information regarding an optimal solution in the initialization of the algorithm. Our main result is that the expected number of iterations before termination with an exact optimal solution is O(n ln(n)). Keywords: Linear Programming, AverageCase Behavior, InfeasibleInteriorPoint Algorithm. Running Title: Probabilistic Analysis of an LP Algorithm 1 Dept. of Management Sciences, University of Iowa. Supported by an Interdisciplinary Research Grant from the Center for Advanced Studies, University of Iowa. 2 Dept. of Mathematics, Valdosta State University. Supported by an Interdisciplinary Research Grant from the Center for Advanced Studies, University of Iowa. 3 Dept. of Mathematics, University of Iowa. Supported by ...
Predictorcorrector methods for sufficient linear complementarity problems in a wide neighborhood of the central path
 Optimization Methods and Software
"... Abstract. A higher order correctorpredictor interiorpoint method is proposed for solving sufficient linear complementarity problems. The algorithm produces a sequence of iterates in the N − ∞ neighborhood of the central path. The algorithm does not depend on the handicap κ of the problem. It has O ..."
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Cited by 14 (6 self)
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Abstract. A higher order correctorpredictor interiorpoint method is proposed for solving sufficient linear complementarity problems. The algorithm produces a sequence of iterates in the N − ∞ neighborhood of the central path. The algorithm does not depend on the handicap κ of the problem. It has O((1 + κ) √ nL) iteration complexity and is superlinearly convergent even for degenerate problems. Key words. neighborhood linear complementarity, interiorpoint, pathfollowing, correctorpredictor, wide AMS subject classifications. 90C51, 90C33 1. Introduction. The MTY predictorcorrector algorithm proposed by Mizuno, Todd and Ye [9] is a typical representative of a large class of MTY type predictorcorrector methods, which play a very important role among primaldual interior point methods. It was the first algorithm for linear programming (LP) that had both polynomial complexity and superlinear convergence. This result was extended to monotone