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Global optimization of nonlinear fractional programming problems in engineering design

ENGINEERING OPTIMIZATION 2005年第4期37卷 399-409页

作者： Tsai, JF Natl Taiwan Univ Technol Dept Business Management Taipei 106 Taiwan

This study proposes a novel method to solve nonlinear fractional programming (NFP) problems occurring frequently in engineering design and management. Fractional terms composed of signomial functions are first decomposed into convex and concave terms by convexification strategies. Then the piecewise linearization technique is used to approximate the concave terms. The NFP program is then converted into a convex program. A global optimum of the fractional program can finally be found within a tolerable error. When compared with most of the current fractional programming methods, which can only treat a problem with linear functions or a single quotient term, the proposed method can solve a more general fractional programming program with nonlinear functions and multiple quotient terms. Numerical examples are presented to demonstrate the usefulness of the proposed method.

关键词： nonlinear fractional programming convexification convex program

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PROBING THE PARETO FRONTIER FOR BASIS PURSUIT SOLUTIONS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2008年第2期31卷 890-912页

作者： van den Berg, Ewout Friedlander, Michael P. Univ British Columbia Dept Comp Sci Vancouver BC V6T 1Z4 Canada

The basis pursuit problem seeks a minimum one-norm solution of an underdetermined least-squares problem. Basis pursuit denoise (BPDN) fits the least-squares problem only approximately, and a single parameter determines a curve that traces the optimal trade-off between the least-squares fit and the one-norm of the solution. We prove that this curve is convex and continuously differentiable over all points of interest, and show that it gives an explicit relationship to two other optimization problems closely related to BPDN. We describe a root-finding algorithm for finding arbitrary points on this curve;the algorithm is suitable for problems that are large scale and for those that are in the complex domain. At each iteration, a spectral gradient-projection method approximately minimizes a least-squares problem with an explicit one-norm constraint. Only matrix-vector operations are required. The primal-dual solution of this problem gives function and derivative information needed for the root-finding method. Numerical experiments on a comprehensive set of test problems demonstrate that the method scales well to large problems.

关键词： basis pursuit convex program duality root-finding Newton's method projected gradient one-norm regularization sparse solutions

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DUAL ASCENT METHODS FOR PROBLEMS WITH STRICTLY convex COSTS AND LINEAR CONSTRAINTS - A UNIFIED APPROACH

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 1990年第1期28卷 214-242页

作者： TSENG, P Massachusetts Inst of Technology MA United States

Consider problems of the form \[({\text{P}})\qquad min \{ . {f(x)} |Ex \geqq b\} ,\] where f is a strictly convex (possibly nondifferentiable) function and E and b are, respectively, a matrix and a vector. A popular method for solving special cases of (P) (e.g., network flow, entropy maximization, quadratic program) is to dualize the constraints $Ex \geqq b$ to obtain a differentiable maximization problem and then apply an iterative ascent method to solve it. This method is simple and can exploit sparsity, thus making it ideal for large-scale optimization and, in certain cases, for parallel computation. Despite its simplicity, however, convergence of this method has been shown only under certain very restrictive conditions and only for certain special cases of (P). In this paper a block coordinate ascent method is presented for solving (P) that contains as special cases both dual coordinate ascent methods and dual gradient methods. It is shown, under certain mild assumptions on f and (P), that this method converges. Also the line searches are allowed to be inexact and, when f is separable, can be done in parallel.

关键词： 49 90 block coordinate ascent strict convexity convex program

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SPARSE OPTIMIZATION WITH LEAST-SQUARES CONSTRAINTS

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SIAM JOURNAL ON OPTIMIZATION 2011年第4期21卷 1201-1229页

作者： van den Berg, Ewout Friedlander, Michael P. Univ British Columbia Dept Comp Sci Vancouver BC V6T 1Z4 Canada

The use of convex optimization for the recovery of sparse signals from incomplete or compressed data is now common practice. Motivated by the success of basis pursuit in recovering sparse vectors, new formulations have been proposed that take advantage of different types of sparsity. In this paper we propose an efficient algorithm for solving a general class of sparsifying formulations. For several common types of sparsity we provide applications, along with details on how to apply the algorithm, and experimental results.

关键词： basis pursuit compressed sensing convex program duality group sparsity matrix completion Newton's method root-finding sparse solutions

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IMPROVED POINTWISE ITERATION-COMPLEXITY OF A REGULARIZED ADMM AND OF A REGULARIZED NON-EUCLIDEAN HPE FRAMEWORK

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SIAM JOURNAL ON OPTIMIZATION 2017年第1期27卷 379-407页

作者： Goncalves, Max L. N. Melo, Jefferson G. Monteiro, Renato D. C. Univ Fed Goias Inst Math & Stat Campus 2Caixa Postal 131 BR-74001970 Goiania Go Brazil Georgia Inst Technol Sch Ind & Syst Engn Atlanta GA 30332 USA

This paper describes a regularized variant of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex programs. It is shown that the pointwise iteration-complexity of the new variant is better than the corresponding one for the standard ADMM method and that, up to a logarithmic term, is identical to the ergodic iteration-complexity of the latter method. Our analysis is based on first presenting and establishing the pointwise iteration-complexity of a regularized non-Euclidean hybrid proximal extragradient framework whose error condition at each iteration includes both a relative error and a summable error. It is then shown that the new ADMM variant is a special instance of the latter framework where the sequence of summable errors is identically zero when the ADMM stepsize is less than one or a nontrivial sequence when the stepsize is in the interval [1, (1 + root 5)/2).

关键词： alternating direction method of multipliers hybrid proximal extragradient method non-Euclidean Bregman distances convex program pointwise iteration-complexity first-order methods inexact proximal point method regularization

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Large deviations rate function for polling systems

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QUEUEING SYSTEMS 2002年第1-2期41卷 13-44页

作者： Delcoigne, F de la Fortelle, A Univ Paris 10 UFR SEGMI F-92000 Nanterre France Inst Natl Rech Informat & Automat F-78153 Le Chesnay France

In this paper, we identify the local rate function governing the sample path large deviation principle for a rescaled process n(-1)Q(nt), where Q(t) represents the joint number of clients at time t in a polling system with N nodes, one server and Markovian routing. By the way, the large deviation principle is proved and the rate function is shown to have the form conjectured by Dupuis and Ellis. We introduce a so called empirical generator consisting of Q(t) and of two empirical measures associated with S-t, the position of the server at time t. One of the main step is to derive large deviations bounds for a localized version of the empirical generator. The analysis relies on a suitable change of measure and on a representation of fluid limits for polling systems. Finally, the rate function is solution of a meaningful convex program. The method seems to have a wide range of application including the famous Jackson networks, as shown at the end of this study. An example illustrates how this technique can be used to estimate stationary probability decay rate.

关键词： large deviations local rate function polling system fluid limits empirical generator change of measure contraction principle entropy convex program

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Geometric dual formulation for first-derivative-based univariate cubic L₁ splines

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JOURNAL OF GLOBAL OPTIMIZATION 2008年第4期40卷 589-621页

作者： Zhao, Y. B. Fang, S. -C. Lavery, J. E. N Carolina State Univ Raleigh NC 27695 USA Chinese Acad Sci Inst Appl Math AMSS Beijing 100080 Peoples R China Tsinghua Univ Dept Math Sci & Ind Engn Beijing 100084 Peoples R China USA Res Off Res Lab Div Math Res Triangle Pk NC 27709 USA

With the objective of generating "shape-preserving" smooth interpolating curves that represent data with abrupt changes in magnitude and/or knot spacing, we study a class of first-derivative-based C-1-smooth univariate cubic L-1 splines. An L-1 spline minimizes the L-1 norm of the difference between the first-order derivative of the spline and the local divided difference of the data. Calculating the coefficients of an L-1 spline is a nonsmooth non-linear convex program. Via Fenchel's conjugate transformation, the geometric dual program is a smooth convex program with a linear objective function and convex cubic constraints. The dual-to-primal transformation is accomplished by solving a linear program.

关键词： conjugate function convex program cubic L-1 spline shape-preserving interpolation piecewise polynomial

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A Partially Inexact Proximal Alternating Direction Method of Multipliers and Its Iteration-Complexity Analysis

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2019年第2期182卷 640-666页

作者： Adona, Vando A. Goncalves, Max L. N. Melo, Jefferson G. Univ Fed Goias IME BR-74001970 Goiania Go Brazil

This paper proposes a partially inexact proximal alternating direction method of multipliers for computing approximate solutions of a linearly constrained convex optimization problem. This method allows its first subproblem to be solved inexactly using a relative approximate criterion, whereas a proximal term is added to its second subproblem in order to simplify it. A stepsize parameter is included in the updating rule of the Lagrangian multiplier to improve its computational performance. Pointwise and ergodic iteration-complexity bounds for the proposed method are established. To the best of our knowledge, this is the first time that complexity results for an inexact alternating direction method of multipliers with relative error criteria have been analyzed. Some preliminary numerical experiments are reported to illustrate the advantages of the new method.

关键词： Alternating direction method of multipliers Relative error criterion Hybrid extragradient method convex program Pointwise iteration-complexity Ergodic iteration-complexity

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An inexact version of the symmetric proximal ADMM for solving separable convex optimization

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NUMERICAL ALGORITHMS 2023年第1期94卷 1-28页

作者： Adona, Vando A. Goncalves, Max L. N. Univ Fed Goias IME Campus 2131 BR-74001970 Goiania GO Brazil

In this paper, we propose and analyze an inexact version of the symmetric proximal alternating direction method of multipliers (ADMM) for solving linearly constrained optimization problems. Basically, the method allows its first subproblem to be solved inexactly in such way that a relative approximate criterion is satisfied. In terms of root the iteration number k, we establish global O(1/ k) pointwise and O(1/k) ergodic convergence rates of the method for a domain of the acceleration parameters, which is consistent with the largest known one in the exact case. Since the symmetric proximal ADMM can be seen as a class of ADMM variants, the new algorithm as well as its convergence rates generalize, in particular, many others in the literature. Numerical experiments illustrating the practical advantages of the method are reported. To the best of our knowledge, this work is the first one to study an inexact version of the symmetric proximal ADMM.

关键词： Symmetric alternating direction method of multipliers convex program Relative error criterion Pointwise iteration-complexity Ergodic iteration-complexity

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EXPLICIT DUALITY FOR convex HOMOGENEOUS programS

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MATHEMATICAL programMING 1976年第2期10卷 176-191页

作者： GLASSEY, CR UNIV CALIF BERKELEY BERKELEYCA 94720

When all the functions that define a convex program are positively homogeneous, then a dual convex program can be constructed which is defined in terms of the primal data only (the primal variables do not appear). Furthermore, the dualizing process, carried out on the dual program, yields the primal. Several well-known examples of convex programs with explicit duals are shown to be special cases.

关键词： Mathematical Method Primal Data Primal Variable convex program Dualizing Process

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