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Generalized Conjugate Gradient Methods for l₁ Regularized convex quadratic programming with Finite Convergence

MATHEMATICS OF OPERATIONS RESEARCH 2018年第1期43卷 275-303页

作者： Lu, Zhaosong Chen, Xiaojun Simon Fraser Univ Dept Math Burnaby BC V5A 1S6 Canada Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China

The conjugate gradient (CG) method is an efficient iterative method for solving large-scale strongly convex quadratic programming (QP). In this paper, we propose some generalized CG (GCG) methods for solving the l(1)-regularized (possibly not strongly) convex QP that terminate at an optimal solution in a finite number of iterations. At each iteration, our methods first identify a face of an orthant and then either perform an exact line search along the direction of the negative projected minimum-norm subgradient of the objective function or execute a CG subroutine that conducts a sequence of CG iterations until a CG iterate crosses the boundary of this face or an approximate minimizer of over this face or a subface is found. We determine which type of step should be taken by comparing the magnitude of some components of the minimum-norm subgradient of the objective function to that of its rest components. Our analysis on finite convergence of these methods makes use of an error bound result and some key properties of the aforementioned exact line search and the CG subroutine. We also show that the proposed methods are capable of finding an approximate solution of the problem by allowing some inexactness on the execution of the CG subroutine. The overall arithmetic operation cost of our GCG methods for finding an epsilon-optimal solution depends on epsilon in O (log(1/epsilon)), which is superior to the accelerated proximal gradient method (Beck and Teboulle [Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2(1):183-202], Nesterov [Nesterov Yu (2013) Gradient methods for minimizing composite functions. Math. Program. 140(1):125-161]) that depends on epsilon in O (1/root epsilon). In addition, our GCG methods can be extended straight-forwardly to solve box-constrained convex QP with finite convergence. Numerical results demonstrate that our methods are very favorable for solving ill-condition

关键词： conjugate gradient method convex quadratic programming l(1)-regularization sparse optimization finite convergence

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A convex quadratic programming model for unit commitment global optimization

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IEEJ TRANSACTIONS ON ELECTRICAL AND ELECTRONIC ENGINEERING 2018年第3期13卷 417-431页

作者： Hou, Wenting Wei, Hua Tan, Jiancheng Lin, Chunmei Guangxi Univ Guangxi Key Lab Power Syst Optimizat & Energy Tec Nanning 530004 Guangxi Zhuang Peoples R China

This paper presents a convex quadratic programming (CQP) model of the thermal unit commitment (UC) problem based on the recent advancement in mathematics. The proposed model employs convex transformation techniques and is able to achieve the global optimal solution. In the CQP model, the startup cost is represented by using two kinds of binary variables (0-1);the nonconvex constraints, namely the minimum up and down times, are expressed as equivalent linear constraints via a set of linear inequalities;then the nonconvex UC problem is transformed into a convex problem with a quadratic objective function and linear constraints. Comparison studies are carried out based on the results obtained from 20 algorithms applied to 10-60unit 24-h systems. The appendices give the details of these results. The CQP model shows unbeatable performance on global optimal searching and arrives at the overall global optimization solution constantly, and so, for the first time, is able to get the globally optimal solution of the UC problem. The proposed CQP formulation serves as a reliable standard reference for various optimization algorithms. (c) 2017 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

关键词： convex quadratic programming equivalent transformation unit commitment global optimization

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Interval convex quadratic programming problems in a general form

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CENTRAL EUROPEAN JOURNAL OF OPERATIONS RESEARCH 2017年第3期25卷 725-737页

作者： Hladik, Milan Charles Univ Prague Fac Math & Phys Dept Appl Math Malostranske Nam 25 Prague 11800 Czech Republic

This paper addresses the problem of computing the minimal and the maximal optimal value of a convex quadratic programming (CQP) problem when the coefficients are subject to perturbations in given intervals. Contrary to the previous results concerning on some special forms of CQP only, we present a unified method to deal with interval CQP problems. The problem can be formulated by using equation, inequalities or both, and by using sign-restricted variables or sign-unrestricted variables or both. We propose simple formulas for calculating the minimal and maximal optimal values. Due to NP-hardness of the problem, the formulas are exponential with respect to some characteristics. On the other hand, there are large sub-classes of problems that are polynomially solvable. For the general intractable case we propose an approximation algorithm. We illustrate our approach by a geometric problem of determining the distance of uncertain polytopes. Eventually, we extend our results to quadratically constrained CQP, and state some open problems.

关键词： convex quadratic programming Interval analysis Uncertainty modeling

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Stability and Accuracy of Inexact Interior Point Methods for convex quadratic programming

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2017年第2期175卷 450-477页

作者： Morini, Benedetta Simoncini, Valeria Univ Florence Florence Italy Univ Bologna Bologna Italy IMATI CNR Pavia Italy

We consider primal-dual interior point methods where the linear system arising at each iteration is formulated in the reduced (augmented) form and solved approximately. Focusing on the iterates close to a solution, we analyze the accuracy of the so-called inexact step, i.e., the step that solves the unreduced system, when combining the effects of both different levels of accuracy in the inexact computation and different processes for retrieving the step after block elimination. Our analysis is general and includes as special cases sources of inexactness due either to roundoff and computational errors or to the iterative solution of the augmented system using typical procedures. In the roundoff case, we recover and extend some known results.

关键词： convex quadratic programming Primal-dual interior point methods Inexact interior point steps

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A Wide Neighborhood Arc-Search Interior-Point Algorithm for convex quadratic programming

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Wuhan University Journal of Natural Sciences 2017年第6期22卷 465-471页

作者： YUAN Beibei ZHANG Mingwang HUANG Zhengwei College of Science China Three Gorges University Yichang 443002 Hubei China College of Economics and Management China Three Gorges University Yichang 443002 Hubei China

In this paper, we propose an arc-search interior-point algorithm for convex quadratic programming with a wide neighborhood of the central path, which searches the optimizers along the ellipses that approximate the entire central path. The favorable polynomial complexity bound of the algorithm is obtained, namely O（nlog（（ x^0）~TS^0/ε）） which is as good as the linear programming analogue. Finally, the numerical experiments show that the proposed algorithm is efficient.

关键词： arc-search interior-point algorithm polynomial complexity convex quadratic programming

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Primal and dual active-set methods for convex quadratic programming

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MATHEMATICAL programming 2016年第1-2期159卷 469-508页

作者： Forsgren, Anders Gill, Philip E. Wong, Elizabeth KTH Royal Inst Technol Dept Math Optimizat & Syst Theory S-10044 Stockholm Sweden Univ Calif San Diego Dept Math La Jolla CA 92093 USA

Computational methods are proposed for solving a convex quadratic program (QP). Active-set methods are defined for a particular primal and dual formulation of a QP with general equality constraints and simple lower bounds on the variables. In the first part of the paper, two methods are proposed, one primal and one dual. These methods generate a sequence of iterates that are feasible with respect to the equality constraints associated with the optimality conditions of the primal-dual form. The primal method maintains feasibility of the primal inequalities while driving the infeasibilities of the dual inequalities to zero. The dual method maintains feasibility of the dual inequalities while moving to satisfy the primal inequalities. In each of these methods, the search directions satisfy a KKT system of equations formed from Hessian and constraint components associated with an appropriate column basis. The composition of the basis is specified by an active-set strategy that guarantees the nonsingularity of each set of KKT equations. Each of the proposed methods is a conventional active-set method in the sense that an initial primal- or dual-feasible point is required. In the second part of the paper, it is shown how the quadratic program may be solved as a coupled pair of primal and dual quadratic programs created from the original by simultaneously shifting the simple-bound constraints and adding a penalty term to the objective function. Any conventional column basis may be made optimal for such a primal-dual pair of shifted-penalized problems. The shifts are then updated using the solution of either the primal or the dual shifted problem. An obvious application of this approach is to solve a shifted dual QP to define an initial feasible point for the primal (or vice versa). The computational performance of each of the proposed methods is evaluated on a set of convex problems from the CUTEst test collection.

关键词： quadratic programming Active-set methods convex quadratic programming Primal active-set methods Dual active-set methods

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Incorporating higher-order point distribution model priors into MRFs using convex quadratic programming

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MACHINE VISION AND APPLICATIONS 2016年第6期27卷 821-832页

作者： Arashloo, Shervin Rahimzadeh Tarbiat Modares Univ Dept Med Informat Fac Med Sci Tehran Iran

Recently, with the advent of powerful optimisation algorithms for Markov random fields (MRFs), priors of high arity (more than two) have been put into practice more widely. The statistical relationship between object parts encoding shape in a covariant space, also known as the point distribution model (PDM), is a widely employed technique in computer vision which has been largely overlooked in the context of higher-order MRF models. This paper focuses on such higher-order statistical shape priors and illustrates that in a spatial transformation invariant space, these models can be formulated as convex quadratic programmes. As such, the associated energy of a PDM may be optimised efficiently using a variety of different dedicated algorithms. Moreover, it is shown that such an approach in the context of graph matching can be utilised to incorporate both a global rigid and a non-rigid deformation prior into the problem in a parametric form, a problem which has been rarely addressed in the literature. The paper then illustrates an application of PDM priors for different tasks using graphical models incorporating factors of different cardinalities.

关键词： MRFs Higher-order potentials Point distribution models convex quadratic programming

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A novel neural network for solving convex quadratic programming problems subject to equality and inequality constraints

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NEUROCOMPUTING 2016年 214卷 23-31页

作者： Huang, Xinjian Lou, Yang Cui, Baotong Jiangnan Univ Key Lab Adv Proc Control Light Ind Minist Educ Wuxi 214122 Peoples R China Jiangnan Univ Sch IoT Engn Wuxi 214122 Peoples R China

This paper proposes a neural network model for solving convex quadratic programming (CQP) problems, whose equilibrium points coincide with Karush-Kuhn-Tucker (KKT) points of the CQP problem. Using the equality transformation and Fischer-Burmeister (FB) function, we construct the neural network model and present the KKT condition for the CQP problem. In contrast to two existing neural networks for solving such problems, the proposed neural network has fewer variables and neurons, which makes circuit realization easier. Moreover, the proposed neural network is asymptotically stable in the sense of Lyapunov such that it converges to an exact optimal solution of the CQP problem. Simulation results are provided to show the feasibility and efficiency of the proposed network. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Neural network convex quadratic programming Fischer-Burmeister function Stability

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On the Solution Existence of convex quadratic programming Problems in Hilbert Spaces

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TAIWANESE JOURNAL OF MATHEMATICS 2016年第6期20卷 1417-1436页

作者： Vu Van Dong Nguyen Nang Tam Phuc Yen Coll Ind Hai Phong City Cam Pha Town Vietnam Hanoi Pedag Inst 2 Hanoi Vietnam

We provide solution existence results for the convex quadratic programming problems in Hilbert spaces, which the constraint set is defined by finitely many convex quadratic inequalities. In order to obtain our results, we shall use either the properties of the Legendre form or the properties of the finite-rank operator. The existence results are established without requesting neither coercivity of the objective function nor compactness of the constraint set.

关键词： convex quadratic programming Solution existence Legendre form Recession cone Hilbert spaces Finite-rank operator

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An exterior point polynomial-time algorithm for convex quadratic programming

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2015年第1期61卷 51-78页

作者： Tian, Da Gang Shanghai Univ Sci & Technol Sch Business Shanghai Peoples R China

In this paper an exterior point polynomial time algorithm for convex quadratic programming problems is proposed. We convert a convex quadratic program into an unconstrained convex program problem with a self-concordant objective function. We show that, only with duality, the Path-following method is valid. The computational complexity analysis of the algorithm is given.

关键词： Self-concordant function Polynomial-time algorithm Exterior point Path-following convex quadratic programming Regularization

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