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The positiveness of lower limits of the Hoffman constant in parametric polyhedral programs

JOURNAL OF GLOBAL OPTIMIZATION 2012年第4期53卷 641-661页

作者： Jourani, A. Zagrodny, D. Polish Acad Sci Syst Res Inst PL-01447 Warsaw Poland Univ Bourgogne CNRS UFR Sci & Tech Inst Math BourgogneUMR 5584 F-21078 Dijon France Cardinal Stefan Wyszynski Univ Coll Sci Fac Math & Nat Sci PL-01815 Warsaw Poland

If K(t) are sets of admissible solutions in parametric programs then it is natural to ask about the Lipschitz-like property and the lower semi-continuity of the multifunction. Answers to this question are related to the problem of the continuity or Lipschitz continuity of the value function, namely having the lower semi-continuity of K(center dot) we get the upper semi-continuity of the function easily and the Lipschitz-like property of K(center dot) leads to the Lipschitz-continuity of it. Herein sufficient conditions to get these properties of the polyhedral multifunction of admissible solutions are given in terms of the lower limit of the Hoffman constant. It is shown that the multifunction is Lipschitz-like at these parameters at which the lower limit of the Hoffman constant are positive.

关键词： parametric programming Hoffman constant Error bound Mosco convergence Attouch's theorem Convex functions Subdifferentials Polyhedrals

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SENSITIVITY ANALYSIS OF PARAMETRIZED PROGRAMS VIA GENERALIZED EQUATIONS

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1994年第2期32卷 553-571页

作者： SHAPIRO, A Georgia Institute of Technology Atlanta GA United States

This paper investigates local behavior of optimal solutions of parametrized optimization problems with cone constraints in Banach spaces. The corresponding first-order optimality conditions are formulated in a form of generalized equations (variational inequalities) and solutions of these generalized equations are studied. It is shown that under certain second-order sufficient optimality conditions and a regularity assumption related to the associated Lagrange multipliers, the considered optimal solutions are Lipschitzian stable. This is compared with a similar result in Shapiro and Bonnans [SIAM J. Control Optim., 30 (1992), pp. 1409-1422]. Under the additional assumption of uniqueness of the Lagrange multipliers, first-order expansions of the optimal solutions are given in terms of solutions of auxiliary optimization problems. Finally, as an example, semi-infinite programming problems are discussed.

关键词： NONLINEAR OPTIMIZATION parametric programming SENSITIVITY ANALYSIS GENERALIZED EQUATIONS LIPSCHITZIAN STABILITY DIRECTIONAL DIFFERENTIABILITY SEMIINFINITE programming

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Semiglobal stabilization of linear systems using constrained controls:: A parametric optimization approach

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INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL 1999年第8期9卷 461-484页

作者： Solís-Daun, J Alvarez-Ramírez, J Suárez, R Univ Autonoma Metropolitana Iztapalapa Div Ciencias Basicas & Ingn Mexico City 09340 DF Mexico

A bounded feedback control for asymptotic stabilization of linear systems is derived. The designed control law increases the feedback gain as the controlled trajectory converges towards the origin. A sequence of invariant sets of decreasing size, associated with a (quadratic) Lyapunov function, are defined and related to each of them, the corresponding possible highest gain is chosen, while maintaining the input bounded. Gains as functions of the position are designed by explicitly solving a c-parameterized programming problem. The proposed method allows global asymptotic stabilization of open-loop stable systems, with inputs subject to magnitude bounds and globally bounded rates. In the general case of linear systems that are asymptotic null controllable with bounded input, the semiglobal stabilization is also addressed taking into account the problem of semiglobal rate-limited actuators. The method is illustrated with the global stabilization of an inertial navigator, and the stabilization of a nonlinear model of a crane with hanging load. Copyright (C) 1999 John Wiley & Sons, Ltd.

关键词： bounded control linear systems semiglobal stabilization parametric programming

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On the consistency of aggregate production frontiers

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2006年第1期172卷 326-333页

作者： Weill, L Univ Robert Schuman Inst Etud Polit F-67082 Strasbourg France

This paper investigates the consistency of efficiency frontier methods applied at the aggregate level. We estimate eight aggregate production frontiers on a sample of 93 countries, depending on three methodological choices for the specification of the frontier: the choice of the approach technique (stochastic frontier approach or data envelopment analysis), the specification of human capital as an input, and the nature of returns to scale. We observe some differences on the descriptive statistics of the distributions of the efficiency scores, but also a very high significant and positive correlation between scores rankings regardless of the methodological choices made. Our results tend then to suggest the consistency of the efficiency techniques at the aggregate level. (C) 2004 Elsevier B.V. All rights reserved.

关键词： data envelopment analysis parametric programming macroeconomic efficiency frontiers

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An approach to parametric nonlinear least square optimization and application to task-level learning control

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 1997年第7期42卷 912-927页

作者： Gorinevsky, D UNIV TORONTO FAC APPL SCI & ENGN TORONTO ON CANADA

This paper considers a parametric nonlinear least square (NLS) optimization problem, Unlike a classical NLS problem statement, we assume that a nonlinear optimized system depends on two arguments: an input vector and a parameter vector, The input vector can be modified to optimize the system, while the parameter vector changes from one optimization iteration to another and is not controlled, The optimization process goal is to find a dependence of the optimal input vector on the parameter vector, where the optimal input vector minimizes a quadratic performance index, The paper proposes an extension of the Levenberg-Marquardt algorithm for a numerical solution of the formulated problem, The proposed algorithm approximates the nonlinear system in a vicinity of the optimum by expanding it into a series of parameter vector functions, affine in the input vector. In particular, a radial basis function network expansion is considered. The convergence proof for the algorithm is presented, The proposed approach is applied to task-level learning control of a two-link flexible arm, Each evaluation of the system in the optimization process means completing a controlled motion of the arm, In the simulation example, the controlled motions take only about 1.5 periods of the lowest eigenfrequency oscillations. The algorithm controls this strongly nonlinear oscillatory system very efficiently, Without any prior knowledge of the system dynamics, it achieves a satisfactory control of arbitrary arm motions after only 500 learning (optimization) iterations.

关键词： approximation convergence flexible manipulator learning control Levenberg-Marquardt optimization parametric nonlinear least squares parametric programming radial basis function network

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A Class of Prediction-Correction Methods for Time-Varying Convex Optimization

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2016年第17期64卷 4576-4591页

作者： Simonetto, Andrea Mokhtari, Aryan Koppel, Alec Leus, Geert Ribeiro, Alejandro Delft Univ Technol Dept Elect Engn Math & Comp Sci NL-2826 CD Delft Netherlands Univ Penn Dept Elect & Syst Engn Philadelphia PA 19104 USA

This paper considers unconstrained convex optimization problems with time-varying objective functions. We propose algorithms with a discrete time-sampling scheme to find and track the solution trajectory based on prediction and correction steps, while sampling the problem data at a constant rate of 1/h, where h is the sampling period. The prediction step is derived by analyzing the iso-residual dynamics of the optimality conditions. The correction step adjusts for the distance between the current prediction and the optimizer at each time step, and consists either of one or multiple gradient steps or Newton steps, which respectively correspond to the gradient trajectory tracking (GTT) or Newton trajectory tracking (NTT) algorithms. Under suitable conditions, we establish that the asymptotic error incurred by both proposed methods behaves asO(h(2)), and in some cases asO(h(4)), which outperforms the state-of-the-art error bound of O(h) for correction-only methods in the gradient-correction step. Moreover, when the characteristics of the objective function variation are not available, we propose approximate gradient and Newton tracking algorithms (AGT and ANT, respectively) that still attain these asymptotical error bounds. Numerical simulations demonstrate the practical utility of the proposed methods and that they improve upon existing techniques by several orders of magnitude.

关键词： Time-varying optimization non-stationary optimization parametric programming prediction-correction methods

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Multiparametric demand transportation problem

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2002年第2期139卷 206-219页

作者： Filippi, C Romanin-Jacur, G Univ Padua Dept Pure & Appl Math I-35131 Padua Italy Univ Padua Dept Management & Engn I-36100 Vicenza Italy

In some applications a minimum cost transportation model arises where supplies are fixed while demands may simultaneously vary. In this paper we analyse the structure of such a model and propose several techniques to describe its behaviour. Our approach is founded on the concept of optimal region, i.e., the subset of demand vectors where a given basic tree is optimal. The proposed algorithm consists in different pivoting strategies designed to: 1. build up a minimal list of basic trees such that the associated optimal regions cover the set of feasible demand vectors;2. analyse the effects of either opening a new supplier or closing an existing one;3. suitably treat the dual degenerate case by building up a minimal representation of every maximal region where the optimal value is linear in the demand vector. Computational complexity is discussed and numerical examples are given. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： transportation network programming parametric programming

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AN ALGORITHM FOR GENERAL RESTRICTED RATIONAL CHEBYSHEV-APPROXIMATION

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SIAM JOURNAL ON NUMERICAL ANALYSIS 1990年第4期27卷 1024-1033页

作者： HETTICH, R ZENCKE, P SAP AG W-6909 WALLDORFGERMANY

In [R. Hettich and P. Zencke, Teubner Studienbücher Mathematik, Leipzig, Stuttgart, 1982] and [G. Speich, Ph.D. thesis, University of Bonn, Bonn, 1981] a Newton-type differential correction algorithm for general rational Chebyshev approximation has been introduced that has been shown to be globally convergent and superlinearly convergent under assumptions weaker than the common condition of unique solutions. Using recent results on parametric semi-infinite programming [Math. programming, 38 (1987), pp. 323–340], it can be shown that all essential assumptions can be dropped without destroying superlinear convergence. Moreover, additional constraints on the problem, such as restrictions on the range, can be treated without destroying the favorable properties of the algorithm.

关键词： 41A20 65D15 90C31 restricted rational Chebyshev approximation differential correction algorithm superlinear convergence parametric programming

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A FAST parametric MAXIMUM FLOW ALGORITHM AND APPLICATIONS

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SIAM JOURNAL ON COMPUTING 1989年第1期18卷 30-55页

作者： GALLO, G GRIGORIADIS, MD TARJAN, RE RUTGERS STATE UNIV DEPT COMP SCINEW BRUNSWICKNJ 08903 PRINCETON UNIV DEPT COMP SCIPRINCETONNJ 08544 AT&T BELL LABS MURRAY HILLNJ 07974

The classical maximum flow problem sometimes occurs in settings in which the arc capacities are not fixed but are functions of a single parameter, and the goal is to find the value of the parameter such that the corresponding maximum flow or minimum cut satisfies some side condition. Finding the desired parameter value requires solving a sequence of related maximum flow problems. In this paper it is shown that the recent maximum flow algorithm of Goldberg and Tarjan can be extended to solve an important class of such parametric maximum flow problems, at the cost of only a constant factor in its worst-case time bound. Faster algorithms for a variety of combinatorial optimization problems follow from the result.

关键词： 68Q20 68Q25 68R05 68R10 05C20 algorithms data structures communication networks complexity flow sharing fractional programming graphs knapsack constraint linear programming maximum flow maximum-density subgraphs network flows network vulnerability networks nonlinear zero-one programming ratio closure record segmentation parallel computations parametric programming provisioning pseudoforest scheduling selection sequencing

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Development of a parametric computer-aided die design system for cold forging

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JOURNAL OF MATERIALS PROCESSING TECHNOLOGY 1999年第1-3期91卷 80-89页

作者： Lee, RS Hsu, QC Su, SL Natl Kaohsiung Inst Technol Dept Mech Engn Kaohsiung 807 Taiwan Natl Cheng Kung Univ Dept Mech Engn Tainan 701 Taiwan

Based on die-design how charts and data bases collected from the literature, the collected flow chart and data bases of die design, the present paper develops a computer-aided die design system using Auto-Lisp. The design characteristics of the die elements and the die assembly have been expressed in parametric form and programmed. The proposed system has an open architecture, therefore, according to the system structure, die-design engineers can extent the die element design data base and programs. So far, three die-design modules have been finalised, namely: forward extrusion;upsetting;and combined extrusion. With the aid of the proposed system, the functions of die element design, die assembly design, automatic graphics and dimensions generation, redesign, dimension constraint correlations and bill of materials will provide efficiency and convenience of die design. (C) 1999 Elsevier Science S.A. All rights reserved.

关键词： cold forging die design parametric programming

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