During this decade, semidefinite programming has emerged as an important area of optimization due to both the success of interiorpointmethods at solving this problem and the growing number of its application areas s...
During this decade, semidefinite programming has emerged as an important area of optimization due to both the success of interiorpointmethods at solving this problem and the growing number of its application areas such as combinatorial optimization, statistics and optimal control. Most of the interiorpointmethods that solve linear programs in polynomial time were readily extended to solve semidefinite programs. At the same time, semidefinite programming relaxations of many hard-to-solve combinatorial optimization problems were found to yield fairly good solutions. A quadratic program with Boolean variables is generally a difficult problem. We establish a theoretical bound for the semidefinite programming relaxation of such problem with additional simple linear constraints. For the special case of the max-cut problem, a nonrandomized rounding procedure for obtaining a feasible solution from an optimal solution of the semidefinite programming relaxation is presented. This nonrandomized procedure is computationally shown to be superior to the famous randomized rounding method of Goemans and Williamson, which produces randomized solutions that only on the average result in an objective value of at least 0.87856 times the optimal value. Also, an alternate semidefinite programming relaxation approach to the general Boolean quadratic program is taken by relaxing an equivalent problem, which itself is derived through semidefinite programming. This alternate relaxing scheme is shown to produce a strictly tighter feasible region, which can only be matched by the conventional method by adding certain cuts. In this dissertation, two interiorpointmethods, originally developed for linearprogramming, are extended to semidefinite programming. Straightforward extension of the affine scaling method to semidefinite programming problems has been shown to fail. We present a slightly modified approach. Numerical experiments show that our algorithm works well on various types of pr
For many years globally convergent probability-one homotopy methods have been remarkably successful on di cult realistic engineering optimization problems, most of which were attacked by homotopy methods because other...
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For many years globally convergent probability-one homotopy methods have been remarkably successful on di cult realistic engineering optimization problems, most of which were attacked by homotopy methods because other optimization algorithms failed or were ineffective. Convergence theory has been derived for a few particular problems, and considerable fixed pointtheory exists, but generally convergence theory for the homotopy maps used in practice for nonlinear constrained optimization has been lacking. This paper derives some probability-one homotopy convergence theorems for unconstrained and inequality constrained optimization, for linear and nonlinear inequality constraints, and with and without convexity. Some insight is provided into why the homotopies used in engineering practice are so successful, and why this success is more than dumb luck. By presenting the theory as variations on a prototype probability-one homotopy convergence theorem, the essence of such convergence theory is elucidated.
Generalized network flow problems generalize normal network Row problems by specifying a Row multiplier mu((v, w)) for each are (v, w). For every unit of Row entering the are, mu((v, w)) units of flow exit. We present...
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Generalized network flow problems generalize normal network Row problems by specifying a Row multiplier mu((v, w)) for each are (v, w). For every unit of Row entering the are, mu((v, w)) units of flow exit. We present a strongly polynomial algorithmn for a single-source generalized shortest paths problem, using a left-distributive: closed semiring. This permits use of a dynamic-programming routine similar to the Bellman-Ford algorithm, given a guess for the value of the optimal solution. Using Megiddo's parametric search scheme, we can compute the optimal value in strongly polynomial time. The algorithm's running time O(mn(2) log n) matches the previously best known, but the algorithm is simpler, is based on the well-known theory of closed semirings, and directly works with the given graph. All previous polynomial-time algorithms were based on interior-pointmethods or directly solved the dual problem and translated the solution back to the primal problem. Using this generalized shortest paths algorithm, we present fully polynomial-time approximation schemes for the generalized versions of the maximum Row, the nonnegative-cost minimum-cost flow, the concurrent how, the multicommodity maximum Row, and the multicommodity nonnegative-cost minimum-cost flow problems with running times independent of the size of the Row multipliers' representation. (C) 2001 Academic Press.
The main aim of this work is to provide some basis for the development of interiorpoint algorithms to minimize piecewise linear objective functions. Specifically, we study a piecewise linear separable and convex obje...
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The main aim of this work is to provide some basis for the development of interiorpoint algorithms to minimize piecewise linear objective functions. Specifically, we study a piecewise linear separable and convex objective function, subject to linear constraints. The available methods in the literature for this class of problem are of the Simplex type, except for specific cases, such as linear fitting in the sense of L-1-norm. A common practice for the resolution of piecewise linear programs consists of transforming them into equivalent linear programs and exploring their structure. This strategy is suitable for simplex-type methods, but inadequate for interiorpointmethods. We show how to extend known interiorpointmethods devised for linearprogramming to piecewise linearprogramming without resorting to equivalent linear programs. We also show that the generated interiorpoints for the original piecewise linear program are not interiorpoints for the equivalent linear program. Finally, some computational experiments are presented.
This paper addresses the local convergence properties of the affine-scaling interior-point algorithm for nonlinearprogramming. The analysis of local convergence is developed in terms of parameters that control the in...
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This paper addresses the local convergence properties of the affine-scaling interior-point algorithm for nonlinearprogramming. The analysis of local convergence is developed in terms of parameters that control the interior-point scheme and the size of the residual of the linear system that provides the step direction. The analysis follows the classical theory for quasi-Newton methods and addresses q-linear, q-superlinear, and q-quadratic rates of convergence.
The modern era of interior-pointmethods dates to 1984, when Karmarkar proposed his algorithm for linearprogramming. In the years since then, algorithms and software for linearprogramming have become quite sophistic...
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The modern era of interior-pointmethods dates to 1984, when Karmarkar proposed his algorithm for linearprogramming. In the years since then, algorithms and software for linearprogramming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semi-definite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. We review some of the key developments in the area, and include comments on the complexity theory and practical algorithms for linearprogramming, semi-definite programming, monotone linear complementarity, and convex programming over sets that can be characterized by self-concordant barrier functions. (C) 2000 Elsevier Science B.V. All rights reserved.
Since Karmarkar's first successful interior-point algorithm for linearprogramming in 1984, the interest and consequently the number of publications in the area have increased tremendously, leaving the newcomers t...
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Since Karmarkar's first successful interior-point algorithm for linearprogramming in 1984, the interest and consequently the number of publications in the area have increased tremendously, leaving the newcomers to the field trapped in a jungle of papers and reports. In this paper we review and classify major publications on interior-pointmethodstheory, on the practical implementation of the most successful interior-point algorithms, and on their applications to power systems optimization problems. A listing of state-of-the-art interior-point software codes and major online research resources in the Internet are included.
Ellipsoids that contain all optimal primal solutions, those that contain all optimal dual slack solutions, and primal-dual ellipsoids are derived. They are independent of the algorithm used and have a smaller size tha...
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Ellipsoids that contain all optimal primal solutions, those that contain all optimal dual slack solutions, and primal-dual ellipsoids are derived. They are independent of the algorithm used and have a smaller size than the Choi-Goldfarb ellipsoids [J. Optim. theory Appl. 80 (1994) 161-173]. (C) 2000 Elsevier Science B.V. All rights reserved.
This paper presents a primal-dual interiorpoint algorithm for linearly constrained convex nonlinearprogramming and computational experience in solving optimal mechanism design problems using the algorithm. These pro...
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This paper presents a primal-dual interiorpoint algorithm for linearly constrained convex nonlinearprogramming and computational experience in solving optimal mechanism design problems using the algorithm. These problems are frequently formulated as convex programming problems, i.e. problems with linear constraints and an objective function formed as a sum of squared quantities, The algorithm has been implemented and tested on an IBM PC computer. The computational results demonstrated that the algorithm finds an approximate optimal solution in fewer iterations and function evaluations, the obtained solution usually being an interior feasible solution, and so the resulting method is very efficient and effective. (C) 1999 Elsevier Science Ltd. All rights reserved.
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