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Primal-Dual Relationship Between Levenberg-Marquardt and Central Trajectories for Linearly Constrained convex Optimization

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2014年第3期162卷 705-717页

作者： Behling, Roger Gonzaga, Clovis Haeser, Gabriel Catolica SC Jaragua Do Sul SC Brazil Univ Fed Santa Catarina Florianopolis SC Brazil Univ Fed Sao Paulo Inst Sci & Technol Sao Jose Dos Campos SP Brazil

We consider the minimization of a convex function on a bounded polyhedron (polytope) represented by linear equality constraints and non-negative variables. We define the Levenberg-Marquardt and central trajectories starting at the analytic center using the same parameter, and show that they satisfy a primal-dual relationship, being close to each other for large values of the parameter. Based on this, we develop an algorithm that starts computing primal-dual feasible points on the Levenberg-Marquardt trajectory and eventually moves to the central path. Our main theorem is particularly relevant in quadratic programming, where points on the primal-dual Levenberg-Marquardt trajectory can be calculated by means of a system of linear equations. We present some computational tests related to box constrained trust region subproblems.

关键词： Central path Levenberg-Marquardt Primal-dual Interior points convex quadratic programming Trust region Initial point

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A branch-and-bound algorithm for instrumental variable quantile regression

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MATHEMATICAL programming COMPUTATION 2017年第4期9卷 471-497页

作者： Xu, Guanglin Burer, Samuel Univ Iowa Dept Management Sci Iowa City IA 52242 USA

This paper studies a statistical problem called instrumental variable quantile regression (IVQR). We model IVQR as a convex quadratic program with complementarity constraints and-although this type of program is generally NP-hard-we develop a branch-and-bound algorithm to solve it globally. We also derive bounds on key variables in the problem, which are valid asymptotically for increasing sample size. We compare our method with two well known global solvers, one of which requires the computed bounds. On random instances, our algorithm performs well in terms of both speed and robustness.

关键词： convex quadratic programming Complementarity constraints Branch-and-bound Quantile regression

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A Numerical Methodology for Enforcing Maximum Principles and the Non-Negative Constraint for Transient Diffusion Equations

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Communications in Computational Physics 2016年第1期19卷 53-93页

作者： K.B.Nakshatrala H.Nagarajan M.Shabouei Department of Civil&Environmental Engineering University of HoustonHoustonTexas 77204-4003USA Department of Mechanial Engineering Texas A&M UniversityCollege StationTX 77843USA.

Transient diffusion equations arise in many branches of engineering and applied sciences(e.g.,heat transfer and mass transfer),and are parabolic partial differential *** is well-known that these equations satisfy important mathematical properties like maximum principles and the non-negative constraint,which have implications in mathematical ***,existing numerical formulations for these types of equations do not,in general,satisfy maximum principles and the nonnegative *** this paper,we present a methodology for enforcing maximum principles and the non-negative constraint for transient anisotropic diffusion *** proposed methodology is based on the method of horizontal lines in which the time is discretized *** results in solving steady anisotropic diffusion equation with decay equation at every discrete *** also present other plausible temporal discretizations,and illustrate their shortcomings in meeting maximum principles and the non-negative *** proposedmethodology can handle general computational grids with no additional restrictions on the *** illustrate the performance and accuracy of the proposed methodology using representative numerical *** also perform a numerical convergence analysis of the proposed *** comparison,we also present the results from the standard singlefield semi-discrete formulation and the results froma popular software package,which all will violate maximum principles and the non-negative constraint.

关键词： Numerical heat and mass transfer maximum principles non-negative solutions anisotropic diffusion method of horizontal lines convex quadratic programming parabolic PDEs

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Embedded Optimization Methods for Industrial Automatic Control

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IFAC-PapersOnLine 2017年第1期50卷 13194-13209页

作者： Ferreau, H.J. Almér, S. Verschueren, R. Diehl, M. Frick, D. Domahidi, A. Jerez, J.L. Stathopoulos, G. Jones, C. ABB Corporate Research Segelhofstrasse 1K Baden-Dättwil5405 Switzerland Systems Control and Optimization Laboratory Department of Microsystems Engineering and Department of Mathematics University of Freiburg Georges-Koehler-Allee 102 Freiburg79110 Germany Automatic Control Laboratory ETH Zürich Physikstrasse 3 Zurich8092 Switzerland embotech GmbH Automatic Control Laboratory Physikstrasse 3 Zurich8092 Switzerland Automatic Control Laboratory École Polytechnique Fédérale de Lausanne Lausanne1015 Switzerland

Starting in the late 1970s, optimization-based control has built up an impressive track record of successful industrial applications, in particular in the petrochemical and process industries. More recently, optimization methods for automatic control are more and more deployed on so-called embedded hardware to cater for application-specific needs such as guaranteed communication latency, low energy consumption or cost effectiveness. This development greatly broadens the scope of applications to which optimization methods can be applied to sectors such as robotics, automotive, aerospace or power electronics. However, it also poses additional challenges regarding both the algorithmic concepts and their actual implementations for a given computing hardware. This survey paper discusses key challenges for using embedded optimization methods and summarizes their main use cases in current industrial practice. Motivated by this discussion, a number of dedicated embedded optimization algorithms and their actual implementations are reviewed. The presentation is organized according to the mathematical structure of the embedded optimization problem, ranging from convex quadratic programming over more general convex and nonconvex problems to formulations comprising discrete optimization variables. © 2017

关键词： Embedded systems Automation Control engineering Cost effectiveness Energy utilization Hardware Interactive computer systems Model predictive control Optimization quadratic programming Real time systems convex quadratic programming Embedded optimizations Industrial controls Low energy consumption Mathematical structure Model based controls Optimal controls Predictive control

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CONTAINING AND SHRINKING ELLIPSOIDS IN THE PATH-FOLLOWING ALGORITHM

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MATHEMATICAL programming 1990年第1期47卷 1-9页

作者： YE, YY TODD, MJ UNIV IOWA DEPT MANAGEMENT SCIIOWA CITYIA CORNELL UNIV DEPT OPERAT RES & IND ENGNITHACANY 14853

We describe a new potential function and a sequence of ellipsoids in the path-following algorithm for convex quadratic programming. Each ellipsoid in the sequence contains all of the optimal primal and dual slack vectors. Furthermore, the volumes of the ellipsoids shrink at the ratio $$2^{ - \Omega (\sqrt n )} $$ , in comparison to 2−Ω(1) in Karmarkar's algorithm and 2−Ω(1/n) in the ellipsoid method. We also show how to use these ellipsoids to identify the optimal basis in the course of the algorithm for linear programming.

关键词： convex quadratic programming ellipsoid method Karmarkar's algorithm Linear programming path-following algorithm

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Nonlinear optimization and support vector machines

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4OR-A QUARTERLY JOURNAL OF OPERATIONS RESEARCH 2018年第2期16卷 111-149页

作者： Piccialli, Veronica Sciandrone, Marco Univ Roma Tor Vergata Dipartimento Ingn Civile & Ingn Informat Via Politecn I-100133 Rome Italy Univ Florence Dipartimento Ingn Informaz Via Santa Marta 3 I-50139 Florence Italy

Support Vector Machine (SVM) is one of the most important class of machine learning models and algorithms, and has been successfully applied in various fields. Nonlinear optimization plays a crucial role in SVM methodology, both in defining the machine learning models and in designing convergent and efficient algorithms for large-scale training problems. In this paper we present the convex programming problems underlying SVM focusing on supervised binary classification. We analyze the most important and used optimization methods for SVM training problems, and we discuss how the properties of these problems can be incorporated in designing useful algorithms.

关键词： Statistical learning theory Support vector machine convex quadratic programming Wolfe's dual theory Kernel functions Nonlinear optimization methods

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Distributed Load Restoration in Unbalanced Active Distribution Systems

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IEEE TRANSACTIONS ON SMART GRID 2019年第5期10卷 5759-5769页

作者： Nejad, Reza Roofegari Sun, Wei Univ Cent Florida Dept Elect & Comp Engn Orlando FL 32816 USA

The resilience of modern power systems has been constantly threatened by various natural disasters and man-made attacks. Under those threats, fast and automatic restoration actions are critical for restoring impacted systems to normal operating conditions and preventing from additional disastrous consequences. Smart grid technologies, such as distributed energy resources (DERs) and microgrids, provide both opportunities and challenges for distribution system restoration. In this paper, distributed load restoration (DLR) in unbalanced active distribution networks is developed using the alternating direction method of multipliers (ADMMs) algorithm. The step-by-step restoration actions are provided to schedule the output of DERs and distribution system control devices, as well as the power consumption from transmission system restoration. The nonlinearities from three-phase unbalanced power flow and distribution components modeling are relaxed into a convex quadratic programming model. Then, the problem is decomposed into subproblems for each node by applying ADMM-DLR;and solves through each agent by exchanging limited information with neighboring nodes in an iterative procedure. The developed models and algorithms are validated and demonstrated through testing of the IEEE 13-node and 123-node test feeders. Simulation results demonstrate the effectiveness of restoring unbalanced distribution system with the hybrid bottom-up and top-down restoration strategies.

关键词： Alternating direction method of multipliers (ADMM) bidirectional unbalanced power flow convex quadratic programming distributed energy resources load restoration

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An interior point method for general large-scale quadratic programming problems

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ANNALS OF OPERATIONS RESEARCH 1996年第1期62卷 419-437页

作者： Boggs, PT Domich, PD Rogers, JE NATL INST STAND & TECHNOL APPL & COMPUTAT MATH DEPTGAITHERSBURGMD 20899 NATL INST STAND & TECHNOL APPL & COMPUTAT MATH DEPTBOULDERCO 80303

In this paper, we present an interior point algorithm for solving both convex and nonconvex quadratic programs. The method, which is an extension of our interior point work on linear programming problems, efficiently solves a wide class of large-scale problems and forms the basis for a sequential quadratic programming (SQP) solver for general large scale nonlinear programs. The key to the algorithm is a three-dimensional cost improvement subproblem, which is solved at every iteration. We have developed an approximate recentering procedure and a novel, adaptive big-M Phase I procedure that are essential to the success of the algorithm. We describe the basic method along with the recentering and big-M Phase I procedures. Details of the implementation and computational results are also presented.

关键词： big-M Phase I procedure convex quadratic programming interior point methods linear programming method of centers multidirectional search direction nonconvex quadratic programming recentering

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Bard-type methods for the linear complementarity problem with symmetric positive definite matrices

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IMA Journal of Management Mathematics 1988年第1期2卷 51-68页

作者： Judice, J.J. Pires, F.M. Departmento de Mathematica Universidade de Coimbra 3000 coimbra Portugal Departmento de Engenharia Civil Universidade Do Porto 4000 Porto Portugal

In this paper, the authors develop two heuristic Block Bard-type methods for solving linear complementarity problems with P-matrices. Two implementations of these algorithms for sparse symmetric P (positive definite) matrices without special structure are described. A computational study of these methods is also presented, and shows that the heuristic methods are quite efficient for the solution of large-scale linear complementarity problems with such matrices. ©Oxford University Press.

关键词： convex quadratic programming Large-scale problems Liner complementarity problem Sparse matrices

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A New Approach to the Stable Set Problem Based on Ellipsoids

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15th International Conference on Integer programming and Combinatorial Optimization (IPCO)

作者： Giandomenico, Monia Letchford, Adam N. Rossi, Fabrizio Smriglio, Stefano Univ Aquila Dept Comp Sci Laquila Italy Univ Lancaster Dept Management Sci Lancaster England

ISBN: (纸本)9783642208072

A new exact approach to the stable set problem is presented, which attempts to avoids the pitfalls of existing approaches based on linear and semidefinite programming. The method begins by constructing an ellipsoid that contains the stable set polytope and has the property that the upper bound obtained by optimising over it is equal to the Lovasz theta number. This ellipsoid is then used to derive cutting planes, which can be used within a linear programming-based branch-and-cut algorithm. Preliminary computational results indicate that the cutting planes are strong and easy to generate.

关键词： stable set problem semidefinite programming convex quadratic programming cutting planes

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