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Obtaining lower bounds from the progressive hedging algorithm for stochastic mixed-integer programs

MATHEMATICAL PROGRAMMING 2016年第1期157卷 47-67页

作者： Gade, Dinakar Hackebeil, Gabriel Ryan, Sarah M. Watson, Jean-Paul Wets, Roger J. -B. Woodruff, David L. Sabre Holdings Southlake TX 76092 USA Sandia Natl Labs POB 5800 Albuquerque NM 87185 USA Texas A&M Univ College Stn TX 77843 USA Iowa State Univ Ames IA 50011 USA Univ Calif Davis Davis CA 95616 USA

We present a method for computing lower bounds in the progressive hedging algorithm (PHA) for two-stage and multi-stage stochastic mixed-integer programs. Computing lower bounds in the PHA allows one to assess the quality of the solutions generated by the algorithm contemporaneously. The lower bounds can be computed in any iteration of the algorithm by using dual prices that are calculated during execution of the standard PHA. We report computational results on stochastic unit commitment and stochastic server location problem instances, and explore the relationship between key PHA parameters and the quality of the resulting lower bounds.

关键词： Stochastic mixed-integer programming decomposition algorithms Lower bounding

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Computing call-blocking probabilities in LEO satellite networks: The single-orbit case

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY 2002年第2期51卷 332-347页

作者： Zaim, AH Rouskas, GN Perros, HG N Carolina State Univ Dept Comp Sci Raleigh NC 27695 USA

We study the problem of carrying voice calls over a low-earth-orbit satellite network and present an analytical model for computing call-blocking probabilities for a single orbit of a satellite constellation. We have devised a method to solve the corresponding Markov process efficiently for orbits of up to five satellites. For orbits consisting of a larger number of satellites, we have developed an approximate decomposition algorithm to compute the call-blocking probabilities by decomposing the system into smaller subsystems and iteratively solving each subsystem in isolation using the exact Markov process. Our approach can capture blocking due to handoffs for both satellite-fixed and earth-fixed constellations. Numerical results demonstrate that our method is accurate for a wide range of traffic patterns and for orbits with a number of satellites that is representative of commercial satellite systems.

关键词： call blocking probability decomposition algorithms handoffs low-earth-orbit (LEO) satellite networks

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An inexact column-and-constraint generation method to solve two-stage robust optimization problems

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OPERATIONS RESEARCH LETTERS 2023年第1期51卷 92-98页

作者： Tsang, Man Yiu Shehadeh, Karmel S. Curtis, Frank E. Lehigh Univ Dept Ind & Syst Engn Bethlehem PA 18015 USA

We propose a new inexact column-and-constraint generation (i-C&CG) method to solve two-stage robust optimization problems. The method allows solutions to the master problems to be inexact, which is desirable when solving large-scale and/or challenging problems. It is equipped with a backtracking routine that controls the trade-off between bound improvement and inexactness. Importantly, this routine allows us to derive theoretical finite convergence guarantees for our i-C&CG method. Numerical experiments demonstrate computational advantages of our i-C&CG method over state-of-the-art column-and-constraint generation methods. (c) 2022 Elsevier B.V. All rights reserved.

关键词： Two-stage stochastic optimization Robust optimization Column-and-constraint generation decomposition algorithms

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SDDP for some interstage dependent risk-averse problems and application to hydro-thermal planning

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2014年第1期57卷 167-203页

作者： Guigues, Vincent Fundacao Getulio Vargas Escola Matemat Aplicada Rio De Janeiro Brazil IMPA Rio De Janeiro Brazil Univ Fed Rio de Janeiro Escola Politecn Dept Engn Ind Ilha Fundao Rio de Janeiro Brazil

We consider interstage dependent stochastic linear programs where both the random right-hand side and the model of the underlying stochastic process have a special structure. Namely, for equality constraints (resp. inequality constraints) the right-hand side is an affine function (resp. a given function b (t) ) of the process value for the current time step t. As for m-th component of the process at time step t, it depends on previous values of the process through a function h (tm) . For this type of problem, to obtain an approximate policy under some assumptions for functions b (t) and h (tm) , we detail a stochastic dual dynamic programming algorithm. Our analysis includes some enhancements of this algorithm such as the definition of a state vector of minimal size, the computation of feasibility cuts without the assumption of relatively complete recourse, as well as efficient formulas for sharing optimality and feasibility cuts between nodes of the same stage. The algorithm is given for both a non-risk-averse and a risk-averse model. We finally provide preliminary results comparing the performances of the recourse functions corresponding to these two models for a real-life application.

关键词： Stochastic programming Risk-averse optimization decomposition algorithms Interstage dependency Monte Carlo sampling

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High-efficient sample point transform algorithm for large-scale complex optimization

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2024年 432卷

作者： Zhou, Caihua Zhao, Haixin Xu, Shengli Dalian Univ Technol Dept Engn Mech Dalian 116023 Peoples R China Dalian Univ Technol Sch Energy & Power Engn Dalian 116023 Peoples R China

decomposition algorithms and surrogate model methods are frequently employed to address large-scale, intricate optimization challenges. However, the iterative resolution phase inherent to decomposition algorithms can potentially alter the background vector, leading to the repetitive evaluation of samples across disparate iteration cycles. This phenomenon significantly diminishes the computational efficiency of optimization. Accordingly, a novel approach, designated the Sample Point Transformation Algorithm (SPTA), is put forth in this paper as a means of enhancing efficiency through a process of mathematical deduction. The mathematical deduction reveals that the difference between sample points in each iteration loop is a simple function related to the inter-group dependent variables. Consequently, the SPTA method achieves the comprehensive transformation of the sample set by establishing a surrogate model of the difference between the sample sets of two cycles with a limited number of sample points, as opposed to conducting a substantial number of repeated samplings. This SPTA is employed to substitute the most timeconsuming step of direct calculation in the classical optimization process. To validate the calculation efficiency, a series of numerical examples were conducted, demonstrating an improvement of approximately 75 % while maintaining optimal accuracy. This illustrates the advantage of the SPTA in addressing large-scale and complex optimization problems.

关键词： Sample point transformation Large-scale complex optimization decomposition algorithms Surrogate model

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Symmetry group analysis of the shallow water and semi-geostrophic equations

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QUARTERLY JOURNAL OF MECHANICS AND APPLIED MATHEMATICS 2006年第1期59卷 95-123页

作者： Bîla, N Mansfield, EL Clarkson, PA Austrian Acad Sci Johann Radon Inst Computat & Appl Math A-4040 Linz Austria Inst Math Stat & Actuarial Sci Canterbury CT3 1BD Kent England

The two-dimensional shallow water equations and their semi-geostrophic approximation that arise in meteorology and oceanography are analysed from the point of view of symmetry groups theory. A complete classification of their associated classical symmetries, potential symmetries, variational symmetries and conservation laws is found. The semi-geostrophic equations are found to lack conservation of angular momentum. We also show how the particle relabelling symmetry can be used to rewrite the semi-geostrophic equations in such a way that a well-defined formal series solution, smooth only in time, may be carried out. We show that such solutions are in the form of an 'infinite linear cascade'.

关键词： decomposition algorithms DIFFERENTIAL-EQUATIONS CONSERVATION-LAWS ALGEBRA COMPUTATION

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Modelling the electrical activity of skeletal muscle tissue using a multi-domain approach

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BIOMECHANICS AND MODELING IN MECHANOBIOLOGY 2020年第1期19卷 335-349页

作者： Klotz, Thomas Gizzi, Leonardo Yavuz, Utku S. Roehrle, Oliver Inst Modelling & Simulat Biomech Syst Pfaffenwaldring 5a D-70569 Stuttgart Germany Stuttgart Ctr Simulat Sci SimTech Pfaffenwaldring 5a D-70569 Stuttgart Germany Univ Twente Biomed Signals & Syst NL-7500 AE Enschede Netherlands

Electromyography (EMG) can be used to study the behaviour of the motor neurons and thus provides insights into the physiology of the central nervous system. However, due to the high complexity of neuromuscular control, EMG signals are challenging to interpret. While the exact knowledge of the excitation patterns of a specific muscle within an in vivo experimental setting remains elusive, simulations allow to systematically investigate EMG signals in a controlled environment. Within this context, simulations can provide virtual EMG data, which, for example, can be used to validate and optimise signal analysis methods that aim to estimate the relationship between EMG signals and the output of motor neuron pools. However, since existing methods, which are employed to compute EMG signals, exhibit deficiencies with respect to the physical model itself as well as with respect to numerical aspects, we propose a novel homogenised continuum model that closely resolves the electro-physiological behaviour of skeletal muscle tissue. The proposed model is based on an extension of the well-established bidomain model and includes a biophysically detailed description of the electrical activity within the tissue, which is due to the depolarisation of the muscle fibre membranes. In contrast to all other published EMG models, which assume that the electrical potential field for each muscle fibre can be calculated independently, the proposed model assumes that the electrical potential in the muscle fibres is coupled to the electrical potential in the extracellular space. We show that the newly proposed model is able to simulate realistic EMG signals and demonstrate the potential to employ the predicted virtual EMG signal in order to evaluate the goodness of automated decomposition algorithms.

关键词： EMG Biophysical Bidomain Multi-scale decomposition algorithms Motor units

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Analytical solution of the Voznesenskiy problem for stationary and nonstationary linear systems

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JOURNAL OF COMPUTER AND SYSTEMS SCIENCES INTERNATIONAL 1996年第5期34卷 47-59页

作者： Krutko, PD Bauman Technical Univ Russian Federation

An exposition is made of the solution of the problem of decomposing and controlling the perturbed motion of a multiply connected system from the condition that autonomous subsystems with respect to each degree of freedom be asymptotically stable and possess a given spectrum. The decomposition algorithms are obtained in analytical form by the method of the inverse problems of dynamics. It is shown that the structure and the parameters of the algorithms are uniquely determined by the structure and the parameters of a mathematical model of the system being controlled and by the given spectrum of the models of the separate channels. The relationships between the algorithms of decomposition and control are written directly from the equations of motion of the system being decomposed and the standard models, with the aid of which the required dynamics of the autonomous subsystems is assigned. An examination is made of one-level and two-level structures. The solutions found are valid both for stationary and for nonstationary systems.

关键词： Voznesenskiy problem decomposition algorithms compensating feedback inverse problems of dynamics auxiliary controlling function

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On solving two-stage distributionally robust disjunctive programs with a general ambiguity set

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EUROPEAN JOURNAL OF OPERATIONAL RESEARCH 2019年第2期279卷 296-307页

作者： Bansal, Manish Mehrotra, Sanjay Virginia Tech Dept Ind & Syst Engn Blacksburg VA 24061 USA Northwestern Univ Dept Ind Engn & Management Sci Evanston IL 60208 USA

We introduce two-stage distributionally robust disjunctive programs (TSDR-DPs) with disjunctive constraints in both stages and a general ambiguity set for the probability distributions. The TSDR-DPs subsume various classes of two-stage distributionally robust programs where the second stage problems are non-convex programs (such as mixed binary programs, semi-continuous program, nonconvex quadratic programs, separable non-linear programs, etc.). TSDR-DP is an optimization model in which the degree of risk aversion can be chosen by decision makers. It generalizes two-stage stochastic disjunctive program (risk-neutral) and two-stage robust disjunctive program (most-conservative). To our knowledge, the foregoing special cases of TSDR-DPs have not been studied until now. In this paper, we develop decomposition algorithms, which utilize Balas' linear programming equivalent for deterministic disjunctive programs or his sequential convexification approach within L-shaped method, to solve TSDR-DPs. We present sufficient conditions under which our algorithms are finitely convergent. These algorithms generalize the distributionally robust integer L-shaped algorithm of Bansal et al. (SIAM J. on Optimization 28: 2360-2388, 2018) for TSDR mixed binary programs, a subclass of TSDR-DPs. Furthermore, we formulate a semi continuous program (SCP) as a disjunctive program and use our results for TSDR-DPs to solve general two-stage distributionally robust SCPs (TSDR-SCPs) and TSDR-SCP having semi-continuous inflow set in the second stage. Published by Elsevier B.V.

关键词： Stochastic programming Distributionally robust disjunctive programs decomposition algorithms Semi-continuous program Reverse convex program

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Cutting plane methods based on the analytic barrier for minimization of a convex function subject to box-constraints

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OPTIMIZATION METHODS & SOFTWARE 2002年第2期17卷 251-269页

作者： Beer, K Skokov, VA Tech Univ Chemnitz Fak Math D-09107 Chemnitz Germany

We analyze three variants of analytic barrier methods for minimization of a convex function under box-constraints: the classical method of analytic barriers (centres), the proximal method of analytic barriers and the modified proximal method of analytic barriers. We give theoretical and practical complexity estimates of these methods and based on numerical tests outline their dependence on the used control parameters. We consider also the case where the objective function is not given in explicit analytic form and may not be defined everywhere.

关键词： nondifferentiable optimization centre methods cutting plane methods decomposition algorithms

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