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Non-negatively constrained image deblurring with an inexact interior point method

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2009年第1期231卷 236-248页

作者： Bonettini, Silvia Serafini, Thomas Univ Ferrara Dipartimento Matemat I-44100 Ferrara Italy Univ Modena & Reggio Emilia Dipartimento Matemat Modena Italy

Nonlinear image deblurring procedures based on probabilistic considerations have been widely investigated in the literature. This approach leads to model the deblurring problem as a large scale optimization problem, with a nonlinear, convex objective function and non-negativity constraints on the sign of the variables. The interior point methods have shown in the last years to be very reliable in nonlinear programs. In this paper we propose an inexact Newton interior point (IP) algorithm designed for the solution of the deblurring problem. The numerical experience compares the IP method with another state-of-the-art method, the Lucy Richardson algorithm, and shows a significant improvement of the processing time. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Image deblurring Deconvolution methods interior point algorithms Regularization techniques

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CONVERGENCE ANALYSIS OF AN INEXACT FEASIBLE interior point METHOD FOR CONVEX QUADRATIC PROGRAMMING

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SIAM JOURNAL ON OPTIMIZATION 2013年第3期23卷 1510-1527页

作者： Gondzio, Jacek Univ Edinburgh Sch Math Edinburgh Midlothian Scotland Univ Edinburgh Maxwell Inst Math Sci Edinburgh Midlothian Scotland

In this paper we will discuss two variants of an inexact feasible interior point algorithm for convex quadratic programming. We will consider two different neighborhoods: a small one induced by the use of the Euclidean norm which yields a short-step algorithm and a symmetric one induced by the use of the infinity norm which yields a (practical) long-step algorithm. Both algorithms allow for the Newton equation system to be solved inexactly. For both algorithms we will provide conditions for the level of error acceptable in the Newton equation and establish the worst-case complexity results.

关键词： inexact Newton method interior point algorithms linear programming quadratic programming worst-case complexity analysis matrix-free methods

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A New Ai-Zhang Type interior point Algorithm for Sufficient Linear Complementarity Problems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2024年第1期202卷 76-107页

作者： E-Nagy, Marianna Varga, Anita Corvinus Univ Budapest Corvinus Ctr Operat Res F spacing diaeresis ovam ter 8 Fovam Ter 8 H-1093 Budapest Hungary Budapest Univ Technol & Econ Dept Differential Equat Inst Math Muegyet Rkp 3 H-1111 Budapest Hungary

In this paper, we propose a new long-step interior point method for solving sufficient linear complementarity problems. The new algorithm combines two important approaches from the literature: the main ideas of the long-step interior point algorithm introduced by Ai and Zhang and the algebraic equivalent transformation technique proposed by Darvay. Similar to the method of Ai and Zhang, our algorithm also works in a wide neighborhood of the central path and has the best known iteration complexity of short-step variants. However, due to the properties of the applied transforming function in Darvay's technique, the wide neighborhood definition in the analysis depends on the value of the handicap. We implemented not only the theoretical algorithm but a greedy variant of the new method (working in a neighborhood independent of the handicap) in MATLAB and tested its efficiency on both sufficient and non-sufficient problem instances. In addition to presenting our numerical results, we also make some interesting observations regarding the analysis of Ai-Zhang type methods.

关键词： Mathematical programming Linear complementarity problems interior point algorithms Algebraic equivalent transformation technique

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Finding improved local minima of power system optimization problems by interior-point methods

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IEEE TRANSACTIONS ON POWER SYSTEMS 2003年第1期18卷 238-244页

作者： Santos, JR Lora, AT Expósito, AG Ramos, JLM Univ Seville Dept Elect Engn Seville 41092 Spain

This paper presents a simple heuristic technique to deal with multiple local minima in nonconvex, nonlinear power system optimization problems by solving a sequence of interior-point subproblems. Both the real-valued and the mixed-integer cases are separately discussed. The method is then applied to the unit commitment problem and its performance on realistic cases is compared with that of a genetic algorithm (GA).

关键词： genetic algorithms (GAs) global optimization interior point algorithms nonconvex mixed-integer optimization

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interior-point METHODS FOR OPTIMAL-CONTROL OF DISCRETE-TIME-SYSTEMS

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1993年第1期77卷 161-187页

作者： WRIGHT, SJ Argonne National Laboratory Mathematics and Computer Science Division Argonne

We show that recently developed interior point methods for quadratic programming and linear complementarity problems can be put to use in solving discrete-time optimal control problems, with general pointwise constraints on states and controls. We describe interior point algorithms for a discrete-time linear-quadratic regulator problem with mixed state/control constraints and show how they can be efficiently incorporated into an inexact sequential quadratic programming algorithm for nonlinear problems. The key to the efficiency of the interior-point method is the narrow-banded structure of the coefficient matrix which is factorized at each iteration.

关键词： interior point algorithms OPTIMAL CONTROL BANDED LINEAR SYSTEMS

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AN interior point ALGORITHM OF O(ROOT-M-VERTICAL-BAR-LN-EPSILON-VERTICAL-BAR) ITERATIONS FOR C1-CONVEX PROGRAMMING

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MATHEMATICAL PROGRAMMING 1992年第2期57卷 239-257页

作者： SUN, J QI, LQ UNIV NEW S WALES SCH MATH KENSINGTON NSW 2033 AUSTRALIA

We present a theoretical result on a Path-following algorithm for convex programs. The algorithm employs a nonsmooth Newton subroutine. It starts from a near center of a restricted constraint set, performs a partial nonsmooth Newton step in each iteration, and converges to a point whose cost is within epsilon accuracy of the optimal cost in O(square-root m\ ln epsilon\) iterations, where m is the number of constraints in the problem. Unlike other interior point methods, the analyzed algorithm only requires a first-order Lipschitzian condition and a generalized Hessian similarity condition on objective and constraint functions. Therefore, our result indicates the theoretical feasibility of applying interior point methods to certain C1-optimization problems instead of C2-problems. Since the complexity bound is unchanged compared with similar algorithms for C2-convex programming, the result shows that the smoothness of functions may not be a factor affecting the complexity of interior point methods.

关键词： COMPLEXITY OF algorithms CONVERGENCE ANALYSIS interior point algorithms NEWTON METHODS

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A Relaxed interior point Method for Low-Rank Semidefinite Programming Problems with Applications to Matrix Completion

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JOURNAL OF SCIENTIFIC COMPUTING 2021年第2期89卷 1-36页

作者： Bellavia, Stefania Gondzio, Jacek Porcelli, Margherita Univ Firenze Dipartimento Ingn Ind Viale Morgagni 40 I-50134 Florence Italy Univ Edinburgh Sch Math James Clerk Maxwell BldgKings Bldg Edinburgh EH9 3FD Midlothian Scotland Univ Bologna Dipartimento Matemat Piazza Porta San Donato 5 I-40126 Bologna Italy

A new relaxed variant of interior point method for low-rank semidefinite programming problems is proposed in this paper. The method is a step outside of the usual interior point framework. In anticipation to converging to a low-rank primal solution, a special nearly low-rank form of all primal iterates is imposed. To accommodate such a (restrictive) structure, the first order optimality conditions have to be relaxed and are therefore approximated by solving an auxiliary least-squares problem. The relaxed interior point framework opens numerous possibilities how primal and dual approximated Newton directions can be computed. In particular, it admits the application of both the first- and the second-order methods in this context. The convergence of the method is established. A prototype implementation is discussed and encouraging preliminary computational results are reported for solving the SDP-reformulation of matrix-completion problems.

关键词： Semidefinite programming interior point algorithms Low rank Matrix completion problems

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Self-scaled barriers and interior-point methods for convex programming

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MATHEMATICS OF OPERATIONS RESEARCH 1997年第1期22卷 1-42页

作者： Nesterov, YE Todd, MJ CORNELL UNIV SCH OPERAT RES & IND ENGNITHACANY 14853 CORNELL CTR APPL MATH ITHACANY

This paper provides a theoretical foundation for efficient interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled. For such problems we devise long-step and symmetric primal-dual methods. Because of the special properties of these cones and barriers, our algorithms can take steps that go typically a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier.

关键词： convex programming conical form interior point algorithms self-concordant barrier self-scaled cone self-scaled barrier path-following algorithms potential-reduction algorithms

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Maximum loadability of power systems using interior point non-linear optimization methods

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IEEE TRANSACTIONS ON POWER SYSTEMS 1997年第1期12卷 162-169页

作者： Irisarri, GD Wang, X Tong, J Mokhtari, S Siemens Empros Power Systems Control Brooklyn Park MN USA

This paper proposes a non-linear optimization interior point (IP) method for the determination of maximum loadability in a power system. Details of the implementation of pure primal-dual and predictor-corrector primal-dual IP algorithms are presented. It is shown that most of the computational effort of the algorithm is taken by the formation and factorization of the augmented Hessian matrix of the IP algorithm. The size of this matrix can be as large as ten times the number of buses in the system. Comparisons of the two IP implementations with large scale power systems with as many as 4000 buses are presented. It is shown that the IP algorithm constitutes an effective method for the determination of the maximum loadability in a power system.

关键词： power system optimization maximum loadability interior point algorithms voltage stability

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Shape structural optimization with an interior point nonlinear programming algorithm

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STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION 2000年第2期20卷 107-115页

作者： Herskovits, J Dias, G Santos, G Soares, CMM Univ Fed Rio de Janeiro COPPE BR-21945970 Rio De Janeiro Brazil CNPq LNCC Rio De Janeiro Brazil Inst Super Tecn IDMEC Lisbon Portugal

The aim of this paper is to study the implementation of an efficient and reliable technique for shape optimization of solids, based on general nonlinear programming algorithms. We also study the practical behaviour for this kind of applications of a quasi-Newton algorithm, based on the Feasible Direction interior point Method for nonlinear constrained optimization. The optimal shape of the solid is obtained iteratively. At each iteration, a new shape is generated by B-spline curves and a new mesh is automatically generated. The control point coordinates are given by the design variables. Several illustrative two-dimensional examples are solved in a very efficient way. We conclude that the present approach is simple to formulate and to code and that our optimization algorithm is appropriate for this problem.

关键词： shape optimization nonlinear optimization algorithms interior point algorithms

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