The authors consider a family of interior-point algorithms for solving linear programming problems and provide the results of their theoretical substantiation. Subsets of algorithms that have a linear rate of converge...
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The authors consider a family of interior-point algorithms for solving linear programming problems and provide the results of their theoretical substantiation. Subsets of algorithms that have a linear rate of convergence, asymptotically independent of the parameters of the problem being solved, and a subset of algorithms that lead to relatively interiorpoints of the set of optimal solutions are identified. The history of the creation and development of the algorithms is described. New modifications of interior-point algorithms are presented, which contain the previously developed algorithms as a special case.
This paper presents the convergence proof and complexity analysis of an interior-point framework that solves linear programming problems by dynamically selecting and adding relevant inequalities. First, we formulate a...
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This paper presents the convergence proof and complexity analysis of an interior-point framework that solves linear programming problems by dynamically selecting and adding relevant inequalities. First, we formulate a new primal-dual interior-point algorithm for solving linear programmes in non-standard form with equality and inequality constraints. The algorithm uses a primal-dual path-following predictor-corrector short-step interior-point method that starts with a reduced problem without any inequalities and selectively adds a given inequality only if it becomes active on the way to optimality. Second, we prove convergence of this algorithm to an optimal solution at which all inequalities are satisfied regardless of whether they have been added by the algorithm or not. We thus provide a theoretical foundation for similar schemes already used in practice. We also establish conditions under which the complexity of such algorithm is polynomial in the problem dimension and address remaining limitations without these conditions for possible further research.
We study and solve the two-stage stochastic extended second-order cone programming problem. We show that the barrier recourse functions and the composite barrier functions for this optimization problem are self-concor...
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We study and solve the two-stage stochastic extended second-order cone programming problem. We show that the barrier recourse functions and the composite barrier functions for this optimization problem are self-concordant families with respect to barrier parameters. These results are used to develop primal decomposition-based interior-point algorithms. The worst case iteration complexity of the developed algorithms is shown to be the same as that for the short- and long-step primal interioralgorithms applied to the extensive formulation of our problem.
We generalize a primal-dual interior-point algorithm (IPA) proposed recently in (Illes T, Rigo PR, Torok R Unified approach of primal-dual interior-point algo-rithms for a new class of AET functions, 2022) to P-*(x)-h...
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We generalize a primal-dual interior-point algorithm (IPA) proposed recently in (Illes T, Rigo PR, Torok R Unified approach of primal-dual interior-point algo-rithms for a new class of AET functions, 2022) to P-*(x)-horizontal linear complementarity problems (LCPs) over Cartesian product of symmetric cones. The algorithm is based on the algebraic equivalent transformation (AET) technique with a new class of AET functions. The new class is a modification of the class of AET functions proposed in (Illes T, Rigo PR, Torok R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022) where only two conditions are used as opposed to three used in (Illes T, Rigo PR, Torok R Unified approach of primal-dual interior-point algorithms for a new class of AET functions, 2022). Furthermore, the algorithm is a feasible algorithm that uses full Nesterov-Todd steps, hence, no calculation of step-size is necessary. Nevertheless, we prove that the proposed IPA has the iteration bound that matches the best known iteration bound for IPAs solving these types of problems.
In this paper,we present a primal-dual interiorpoint algorithm for semidefinite optimization problems based on a new class of kernel *** functions constitute a combination of the classic kernel function and a barrier...
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In this paper,we present a primal-dual interiorpoint algorithm for semidefinite optimization problems based on a new class of kernel *** functions constitute a combination of the classic kernel function and a barrier *** derive the complexity bounds for large and small-update methods *** show that the best result of iteration bounds for large and small-update methods can be achieved,namely O(q√n(log√n)^q+1/q logn/ε)for large-update methods and O(q^3/2(log√q)^q+1/q√nlogn/ε)for small-update *** test the efficiency and the validity of our algorithm by running some computational tests,then we compare our numerical results with results obtained by algorithms based on different kernel functions.
In the present study, bio-inspired computational heuristics are exploited for finding the solution of economic load dispatch (ELD) problem with valve point loading effect using variants of genetic algorithm (GA) hybri...
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In the present study, bio-inspired computational heuristics are exploited for finding the solution of economic load dispatch (ELD) problem with valve point loading effect using variants of genetic algorithm (GA) hybrid with sequential quadratic programming (SQP) and interior-point algorithms (IPAs). Variants of GAs are constructed using different sets of routines for its fundamental operators in order to explore the entire search space for global optimum solutions while SQP and IPA are integrated with GAs for rapid local convergence. Nine variants of each design scheme based on GAs, GA-SQP and GA-IPAs are applied on three different ELD problems of thermal power plant systems. Comparative studies of the proposed schemes are performed through the results of statistical performance indices in order to establish the worth and effectiveness in terms of accuracy, convergence and complexity measures.
Optimization problems in which a quadratic objective function is optimized subject to linear constraints on the parameters are known as quadratic programming problems (QPs). This focus article reviews algorithms for c...
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Optimization problems in which a quadratic objective function is optimized subject to linear constraints on the parameters are known as quadratic programming problems (QPs). This focus article reviews algorithms for convex QPs (in which the objective is a convex function) and provides pointers to various online resources about QPs. (C) 2015 Wiley Periodicals, Inc.
Kernel functions play an important role in defining new search directions for interior-point algorithms for solving monotone linear complementarity problems. In this paper we present a new kernel function which yields...
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Decomposition techniques for linear programming are difficult to extend to conic optimization problems with general nonpolyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling ...
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Decomposition techniques for linear programming are difficult to extend to conic optimization problems with general nonpolyhedral convex cones because the conic inequalities introduce an additional nonlinear coupling between the variables. However in many applications the convex cones have a partially separable structure that allows them to be characterized in terms of simpler lower-dimensional cones. The most important example is sparse semidefinite programming with a chordal sparsity pattern. Here partial separability derives from the clique decomposition theorems that characterize positive semidefinite and positive-semidefinite-completable matrices with chordal sparsity patterns. The paper describes a decomposition method that exploits partial separability in conic linear optimization. The method is based on Spingarn's method for equality constrained convex optimization, combined with a fast interior-point method for evaluating proximal operators.
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