Solving Obstacle Problems by Using a New Interior Point Algorithm. Abstract
|
|
- Sydney Bailey
- 5 years ago
- Views:
Transcription
1 Solving Obstacle Problems by Using a New Interior Point Algorithm Yi-Chih Hsieh Department of Industrial Engineering National Yunlin Polytechnic Institute Huwei, Yunlin 6308 Taiwan and Dennis L. Bricer Department of Industrial Engineering University of Iowa Iowa City, IA 54 USA (August, 996) Abstract A new infeasible path-following algorithm, which follows a path on the complementarity surface, is proposed for solving obstacle problems. The sequence of iterates generated by the algorithm does not satisfies the primal-dual feasibilities as do the other path-following algorithms which follow the central path, but satisfies the complementarity equations at each iteration. Numerical results show that only few number of iterations are required for solving such class of problems. Keywords: Infeasible Interior Point algorithm, Obstacle Problems, Quadratic Programming.
2 . Introduction Obstacle problems are typical problems in engineering and Physics. Several problems, such as the lubrication problem, the elastoplastic torsion problem and Signorini problem etc., are all the applied obstacle problems (Rodrigues (987)). For obstacle problems, it is assumed that a homogeneous membrane occupying a domain D of the Oxy plane is equally stretched in all direction by a uniform tension and loaded by a normal uniformly distributed force ρ, and it is also assumed that each point (x,y) of the membrane is displaced by an amount v( x, y) vertically to the plane Oxy. One deforms the boundary D of the membrane conformly by prescribing its displacement g = g( x, y), that is, v = g on D. Suppose that the potential energy of the deformed is proportional to the increase in the area of its surface. Since the area is + v x + v D y dxdy + v ( x + v y ) dxdy, D the change in the area of the membrane is equal to v ( x + v ) D y dxdy = D v dd. The potential energy of deformation has the following expression d(v) = λ and without loss of generality, it is set λ =. D v dd, Since the wor done by the external forces during the actual displacement is given by thus the total potential energy is given by e(v) = D ρvdd, d(v) e(v) = D v dd ρv dd. (.) D The obstacle problem is to determine the surface of an elastic membrane subject to a vertical force with upper and lower bounds of the surface. More explicitly speaing, it is to find the
3 equilibrium position, with minimal total potential energy in (.), of an elastic membrane subject to the following restrictions:. the membrane is subject to the action of a vertical force,. the membrane must lie over an "obstacle" (constraints) within the upper and lower bounds. The mathematical formulation for the obstacle problem is expressed by Min q(v) = D v dd ρvdd D s.t. v { v H 0 (D) : l v u on D} (.) where ρ is a force function, D is a bounded open set, and H 0 (D) is a space of all functions with compact support in D such that v and v belong to L (D) (see Ciarlet (978) for details). Several researchers have numerically solved the obstacle problems with various approaches. Moré and Toraldo (99) proposed an algorithm that uses the conjugate gradient method to explore the face of the feasible region defined by the current iterate, and the gradient projection method to move to a different face. The algorithm is limited to problems with bound constraints. Contrarily, Han et al. (99) implement the potential reduction algorithm (an interior point algorithm) to solve the obstacle problem using a given feasible initial solution. In this paper, a new infeasible path-following algorithm in which the initial solution is positive, infeasible, and exactly on the complementarity surface is used to solve the obstacle problem. As numerical results shown, this approach can find solution with the use of only few iterations.. The New Infeasible Interior Point Algorithm In this paper, obstacle problems are discretized into convex quadratic programming and then solved by our new infeasible interior point algorithm. The procedure of this algorithm for general convex quadratic programming is derived in this section, and the convergence of this algorithm can be found in Hsieh and Bricer (996). Consider the convex quadratic programming problems with linear constraints in the following standard form:
4 (QP) Min s.t. xt Qx + c T x Ax y = b x, y 0 Its dual is: (QPD) Max xt Qx + b T w s.t. Qx + A T w + s = c s,w 0 where x, s,c R n, y,w, b R m, Q R n n, and A R m n. The following assumptions are also imposed. (A) The matrix Q is positive semi-definite. (A) The constraint matrix A has full row ran. (A3) The feasible region is nonempty and bounded. For x, y > 0 in (QP) and s, w > 0 in (QPD), we can apply the logarithmic barrier function technique, and obtain the nonlinear programming problems, (QP µ ) and (QPD µ ): n m (QPµ) Min xt Qx + c T x µ log x j µ log y j s.t. Ax y = b x, y > 0 j = j = and (QPDµ) Max xt Qx + b T w + µ log w j + µ log s j s.t. Qx + A T w + s = c w,s > 0 m j = n j = where µ > 0 is a barrier parameter. It is expected that the optimal solution of problem (QP µ ) would converge to the optimal solution of the original problem (QP) as µ 0. Convex programming theory further implies that the global solution, if one exists, is completely characterized by the KKT conditions as: Ax y = b x, y > 0 (primal feasibility) (.a) Qx + A T w + s = c s,w > 0 (dual feasibility) (.b)
5 XSe n = µe n (complementary slacness) (.c) WYe m = µe m (complementary slacness) (.d) where X,S,W, and Y are diagonal matrices with diagonal entries equal to the components of x, s, w, and y, respectively, and e i is the column matrix with i elements, each with value one. Assume that (x,y,s,w ) >0 is a current solution of equation (.) for given µ >0. Let x = e z x, y = e z y, s = e z s, and w = e z w. (.) Substituting (.) to (.) and applying Newton's method, we have AX Y 0 0 QX 0 S A T W X S 0 X S 0 0 W Y 0 W Y dz x dz y dz s dz w η (Ax y b) η (Qx + A T w + s c) = X S e n µ e = n W Y e m µ e m t t t 3 t 4 (.3) where 0 < η <, which is used to control the norm of the direction vector. Therefore, after solving (.3), a new solution can be obtained by step size one (normal step size) such that x + = x e dz x, y + = y e dz y, s + = s e dz s, and w + = w e dz w (.4) For each iteration, the barrier parameter µ is adjusted as follows: µ = σ (x ) T s + (y ) T w n + m, where 0 < σ <. (.5) Next, we introduce the infeasible interior point algorithm for QP problems. Infeasible Interior Point Algorithm. Step :(Initialization) Set =0. Choose three small values for ε, ε, and ε 3, respectively. Define σ 0 and η 0. Start with any initial solution (x 0,y 0,s 0,w 0 )>0 which satisfy X 0 S 0 e n = e n and W 0 Y 0 e m = e m. Step : (Intermediate computation) Compute µ by (.5) and t,t,t 3 and t 4 by (.3), respectively. Step 3 : (Checing optimality) If µ < ε, t b + < ε, and t Qx + c + < ε 3, then stop; the current solution is accepted as the optimal solution. Else proceed to the next step. Step 4 : (Finding the directions) Compute dz w,dz x,dz y, and dz s by (.3).
6 Step 5 : (Finding the new iterate) Compute x +, y +,s + and w + by (.4). Set = + and go to step. It should be noted that the sequence generated by the algorithm follows a path on the complementarity surface, that is, (.c) and (.d) are satisfied for each iteration, which is different from the central path ((.a) and (.b) are satisfied for each iteration) followed by most of infeasible path-following algorithms (Renegar (988), Monteiro and Adler (989a, 989b), and Monteiro, Adler and Resende (990) etc.), with the use of Newton's method for solving (.). One should be also noted that if Monomial method is used to solve (.), the sequence generated follows a similar path on the complementarity surface as that of by our new algorithm (Hsieh and Bricer (996)). 3. Discretization and Numerical Results After finite approximation, the obstacle problem is reduced to a QP problem with box constraints. Let D=[a,a ] [b,b ] be a rectangle in R. To discretize, we draw m x equally spaced vertical lines perpendicular to the x axis between [a,a ] and m y equally spaced vertical lines perpendicular to the y axis between [b,b ]. The grid points formed by the intersection of these lines are labeled to v i, j. For simplicity, assume that the force function ρ is constant. Then the objective function of the obstacle problem (.) becomes q(v) = 4 m x m y q i, j i = j = m x m y ρh x h y v i, j i= j = where v i+, j v i, j q i, j (v) = h x h y h x + v v i, j + i, j h y + v i,j v i, j h x + v v i, j i, j, h y h x = a a m x +, and h y = b b m y +. If we further assume that m x = m y = m and assign a variable x i, j to each grid point v i, j, then the matrix Q for problem (QP) may be expressed as a symmetric positive definite matrix
7 P I I P I 0 I P I Q = R n n, I P I I P where P = R m m, I R m m is the identity matrix, and n = m. The vector c for problem (QP) is given by c i = ρh, i =,,..., n, where h = m +. Hence the obstacle problem in (.) can be discretized by a quadratic programming (QP). In this paper, two obstacle problems with characteristics in Table are solved by the new pathfollowing algorithm. All the numerical results are obtained by using HP75/75 worstation with program coded in APL. Table The characteristics for the obstacle problems. Obstacle Prob. Problem (I) Problem (II) Force ρ = ρ = Lower bounds ( ( ) sin( 9.3γ i )) l i = sin( 3.α i ) sin( 3.3γ i ) l i = sin 9.α i Note: For these two obstacle problems, α i = i i m m for i =,,..., n. ( ) h and γ i = i m h
8 We apply the algorithm to solve these two obstacle problems under the following conditions. (C) n = 65. That is, there are 65 primal variables for the problem, and the total number of variables for the KKT equations is,500. (C) The initial iterate is x 0 = y 0 = (0,...,0) t R n and s 0 = w 0 = (0.0,...,0.0) t R n is infeasible for (.a) and (.b), and is exactly on a path of the complementarity surface. (C3) σ 0 = 0.5. (C4) ε = ε = ε 3 = (C5) η 0 =0.95 and η + = max(0., 0.5η ) if max(dzx i,dzs i, dzy j,dzw j ) > 5 i, j. min(0.95,.η ) otherwise The results are illustrated in Figures -0. From these figures we observe that. For Problem (I), figure of the obstacle based upon the constraints in Table is depicted in Figure, and the initial iterate is depicted in Figure which is a plane, since all initial solutions have value 0. Figure 3, for iteration 5, shows that there is a trend for the plane to project upwards for several points near the boundary. We also find that Figure 4 (at iteration 0) is very similar to Figure 5, the final solution (at iteration 0).. The residuals of the primal-dual feasibility and complementarity for the Problem (I) are illustrated in Figure 6. From this figure we observe that the residual for the complementary slacness is reduced faster than that of primal feasibility and dual feasibility. 3. For Problem (II), we observe the similar property. After 0 iterations, the solution, Figure 7, is extremely close to the final solution (at iteration 5), Figure The total number of iterations needed for these two problems are 0 and 5, respectively, which are very small even though the total number of variables is up to, Conclusion Obstacle problems have been widely investigated and have various applications in the real world. In this paper, after discretization of the obstacle problem into a quadratic programming, a new
9 infeasible interior point algorithm, which follows a path on the complementarity surface, is used to solve such obstacle problems. Limited numerical results show that this new path-following algorithm can efficiently solve large scale quadratic problems with requiring only few iterations. References Ciarlet, P.G. (978) The finite element method elliptic problems. (Studies in Mathematics and Its Application, Vol. 4, edited by Linos, J.L.), North-Holland, NY. Han, C.G., Pardalos, P.M. and Ye, Y. (99) Solving some engineering problems using an interiorpoint algorithm, woring paper, Pennsylvania State University, University Par, PA. Hsieh, Yi-Chih, and Dennis L. Bricer (996) New infeasible interior point algorithm based on Monomial method", Computers & Operations Research, 3, Hsieh, Y.C. and Bricer, D.L. (996) Infeasible interior following a path on the complementarity surface, woring paper (submitted), University of Iowa, IA. Monteiro, R.D.C. and Adler, I. (989a) Interior path following primal-dual algorithms. Part I: linear programming, Mathematical Programming, 44, 7-4. Monteiro, R.D.C. and Adler, I. (989b) Interior path following primal-dual algorithms. Part II: convex quadratic programming, Mathematical Programming, 44, Monteiro, R.D.C., Alder, I. and Resende, M. (990) A polynomial-time primal-dual affine scaling algorithm for linear and convex quadratic programming and its power extension, Mathematics of Operations Research, 5, 9-4. Moré, J.J. and Toraldo, G. (99) On the solution of large quadratic programming problems with bound constraints, SIAM Journal on Optimization,, Rodrigues, J.J. (987) Obstacle problems in mathematical Physics. (North-Holland Mathematics Studies, Vol. 34, edited by Nachbin, L.), North-Holland, NY.
Following The Central Trajectory Using The Monomial Method Rather Than Newton's Method
Following The Central Trajectory Using The Monomial Method Rather Than Newton's Method Yi-Chih Hsieh and Dennis L. Bricer Department of Industrial Engineering The University of Iowa Iowa City, IA 52242
More informationNew Infeasible Interior Point Algorithm Based on Monomial Method
New Infeasible Interior Point Algorithm Based on Monomial Method Yi-Chih Hsieh and Dennis L. Bricer Department of Industrial Engineering The University of Iowa, Iowa City, IA 52242 USA (January, 1995)
More informationOperations Research Lecture 4: Linear Programming Interior Point Method
Operations Research Lecture 4: Linear Programg Interior Point Method Notes taen by Kaiquan Xu@Business School, Nanjing University April 14th 2016 1 The affine scaling algorithm one of the most efficient
More informationLP. Kap. 17: Interior-point methods
LP. Kap. 17: Interior-point methods the simplex algorithm moves along the boundary of the polyhedron P of feasible solutions an alternative is interior-point methods they find a path in the interior of
More informationInterior Point Methods in Mathematical Programming
Interior Point Methods in Mathematical Programming Clóvis C. Gonzaga Federal University of Santa Catarina, Brazil Journées en l honneur de Pierre Huard Paris, novembre 2008 01 00 11 00 000 000 000 000
More informationA FULL-NEWTON STEP INFEASIBLE-INTERIOR-POINT ALGORITHM COMPLEMENTARITY PROBLEMS
Yugoslav Journal of Operations Research 25 (205), Number, 57 72 DOI: 0.2298/YJOR3055034A A FULL-NEWTON STEP INFEASIBLE-INTERIOR-POINT ALGORITHM FOR P (κ)-horizontal LINEAR COMPLEMENTARITY PROBLEMS Soodabeh
More informationInterior-Point Methods
Interior-Point Methods Stephen Wright University of Wisconsin-Madison Simons, Berkeley, August, 2017 Wright (UW-Madison) Interior-Point Methods August 2017 1 / 48 Outline Introduction: Problems and Fundamentals
More informationA Generalized Homogeneous and Self-Dual Algorithm. for Linear Programming. February 1994 (revised December 1994)
A Generalized Homogeneous and Self-Dual Algorithm for Linear Programming Xiaojie Xu Yinyu Ye y February 994 (revised December 994) Abstract: A generalized homogeneous and self-dual (HSD) infeasible-interior-point
More informationAdvanced Mathematical Programming IE417. Lecture 24. Dr. Ted Ralphs
Advanced Mathematical Programming IE417 Lecture 24 Dr. Ted Ralphs IE417 Lecture 24 1 Reading for This Lecture Sections 11.2-11.2 IE417 Lecture 24 2 The Linear Complementarity Problem Given M R p p and
More informationPrimal-dual relationship between Levenberg-Marquardt and central trajectories for linearly constrained convex optimization
Primal-dual relationship between Levenberg-Marquardt and central trajectories for linearly constrained convex optimization Roger Behling a, Clovis Gonzaga b and Gabriel Haeser c March 21, 2013 a Department
More informationCS711008Z Algorithm Design and Analysis
CS711008Z Algorithm Design and Analysis Lecture 8 Linear programming: interior point method Dongbo Bu Institute of Computing Technology Chinese Academy of Sciences, Beijing, China 1 / 31 Outline Brief
More informationLecture 18: Optimization Programming
Fall, 2016 Outline Unconstrained Optimization 1 Unconstrained Optimization 2 Equality-constrained Optimization Inequality-constrained Optimization Mixture-constrained Optimization 3 Quadratic Programming
More information2.098/6.255/ Optimization Methods Practice True/False Questions
2.098/6.255/15.093 Optimization Methods Practice True/False Questions December 11, 2009 Part I For each one of the statements below, state whether it is true or false. Include a 1-3 line supporting sentence
More informationInterior-Point Methods for Linear Optimization
Interior-Point Methods for Linear Optimization Robert M. Freund and Jorge Vera March, 204 c 204 Robert M. Freund and Jorge Vera. All rights reserved. Linear Optimization with a Logarithmic Barrier Function
More informationPrimal-Dual Interior-Point Methods for Linear Programming based on Newton s Method
Primal-Dual Interior-Point Methods for Linear Programming based on Newton s Method Robert M. Freund March, 2004 2004 Massachusetts Institute of Technology. The Problem The logarithmic barrier approach
More informationAn Infeasible Interior Point Method for the Monotone Linear Complementarity Problem
Int. Journal of Math. Analysis, Vol. 1, 2007, no. 17, 841-849 An Infeasible Interior Point Method for the Monotone Linear Complementarity Problem Z. Kebbiche 1 and A. Keraghel Department of Mathematics,
More informationPenalty and Barrier Methods General classical constrained minimization problem minimize f(x) subject to g(x) 0 h(x) =0 Penalty methods are motivated by the desire to use unconstrained optimization techniques
More informationChapter 6 Interior-Point Approach to Linear Programming
Chapter 6 Interior-Point Approach to Linear Programming Objectives: Introduce Basic Ideas of Interior-Point Methods. Motivate further research and applications. Slide#1 Linear Programming Problem Minimize
More informationAdditional Exercises for Introduction to Nonlinear Optimization Amir Beck March 16, 2017
Additional Exercises for Introduction to Nonlinear Optimization Amir Beck March 16, 2017 Chapter 1 - Mathematical Preliminaries 1.1 Let S R n. (a) Suppose that T is an open set satisfying T S. Prove that
More informationConvergence Analysis of Inexact Infeasible Interior Point Method. for Linear Optimization
Convergence Analysis of Inexact Infeasible Interior Point Method for Linear Optimization Ghussoun Al-Jeiroudi Jacek Gondzio School of Mathematics The University of Edinburgh Mayfield Road, Edinburgh EH9
More informationInterior Point Methods for Mathematical Programming
Interior Point Methods for Mathematical Programming Clóvis C. Gonzaga Federal University of Santa Catarina, Florianópolis, Brazil EURO - 2013 Roma Our heroes Cauchy Newton Lagrange Early results Unconstrained
More informationAn interior-point trust-region polynomial algorithm for convex programming
An interior-point trust-region polynomial algorithm for convex programming Ye LU and Ya-xiang YUAN Abstract. An interior-point trust-region algorithm is proposed for minimization of a convex quadratic
More informationMVE165/MMG631 Linear and integer optimization with applications Lecture 13 Overview of nonlinear programming. Ann-Brith Strömberg
MVE165/MMG631 Overview of nonlinear programming Ann-Brith Strömberg 2015 05 21 Areas of applications, examples (Ch. 9.1) Structural optimization Design of aircraft, ships, bridges, etc Decide on the material
More information10 Numerical methods for constrained problems
10 Numerical methods for constrained problems min s.t. f(x) h(x) = 0 (l), g(x) 0 (m), x X The algorithms can be roughly divided the following way: ˆ primal methods: find descent direction keeping inside
More informationA Distributed Newton Method for Network Utility Maximization, I: Algorithm
A Distributed Newton Method for Networ Utility Maximization, I: Algorithm Ermin Wei, Asuman Ozdaglar, and Ali Jadbabaie October 31, 2012 Abstract Most existing wors use dual decomposition and first-order
More informationAn Infeasible Interior-Point Algorithm with full-newton Step for Linear Optimization
An Infeasible Interior-Point Algorithm with full-newton Step for Linear Optimization H. Mansouri M. Zangiabadi Y. Bai C. Roos Department of Mathematical Science, Shahrekord University, P.O. Box 115, Shahrekord,
More informationRESEARCH ARTICLE. A strategy of finding an initial active set for inequality constrained quadratic programming problems
Optimization Methods and Software Vol. 00, No. 00, July 200, 8 RESEARCH ARTICLE A strategy of finding an initial active set for inequality constrained quadratic programming problems Jungho Lee Computer
More informationInterior Point Methods for Convex Quadratic and Convex Nonlinear Programming
School of Mathematics T H E U N I V E R S I T Y O H F E D I N B U R G Interior Point Methods for Convex Quadratic and Convex Nonlinear Programming Jacek Gondzio Email: J.Gondzio@ed.ac.uk URL: http://www.maths.ed.ac.uk/~gondzio
More informationA full-newton step infeasible interior-point algorithm for linear programming based on a kernel function
A full-newton step infeasible interior-point algorithm for linear programming based on a kernel function Zhongyi Liu, Wenyu Sun Abstract This paper proposes an infeasible interior-point algorithm with
More informationResearch Note. A New Infeasible Interior-Point Algorithm with Full Nesterov-Todd Step for Semi-Definite Optimization
Iranian Journal of Operations Research Vol. 4, No. 1, 2013, pp. 88-107 Research Note A New Infeasible Interior-Point Algorithm with Full Nesterov-Todd Step for Semi-Definite Optimization B. Kheirfam We
More informationConjugate Gradient (CG) Method
Conjugate Gradient (CG) Method by K. Ozawa 1 Introduction In the series of this lecture, I will introduce the conjugate gradient method, which solves efficiently large scale sparse linear simultaneous
More informationHomework 5. Convex Optimization /36-725
Homework 5 Convex Optimization 10-725/36-725 Due Tuesday November 22 at 5:30pm submitted to Christoph Dann in Gates 8013 (Remember to a submit separate writeup for each problem, with your name at the top)
More informationA new primal-dual path-following method for convex quadratic programming
Volume 5, N., pp. 97 0, 006 Copyright 006 SBMAC ISSN 00-805 www.scielo.br/cam A new primal-dual path-following method for convex quadratic programming MOHAMED ACHACHE Département de Mathématiques, Faculté
More informationA Regularized Interior-Point Method for Constrained Nonlinear Least Squares
A Regularized Interior-Point Method for Constrained Nonlinear Least Squares XII Brazilian Workshop on Continuous Optimization Abel Soares Siqueira Federal University of Paraná - Curitiba/PR - Brazil Dominique
More informationA PRIMAL-DUAL INTERIOR POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMS. 1. Introduction Consider the quadratic program (PQ) in standard format:
STUDIA UNIV. BABEŞ BOLYAI, INFORMATICA, Volume LVII, Number 1, 01 A PRIMAL-DUAL INTERIOR POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMS MOHAMED ACHACHE AND MOUFIDA GOUTALI Abstract. In this paper, we propose
More informationSolution Methods. Richard Lusby. Department of Management Engineering Technical University of Denmark
Solution Methods Richard Lusby Department of Management Engineering Technical University of Denmark Lecture Overview (jg Unconstrained Several Variables Quadratic Programming Separable Programming SUMT
More informationSVM May 2007 DOE-PI Dianne P. O Leary c 2007
SVM May 2007 DOE-PI Dianne P. O Leary c 2007 1 Speeding the Training of Support Vector Machines and Solution of Quadratic Programs Dianne P. O Leary Computer Science Dept. and Institute for Advanced Computer
More informationHomework 4. Convex Optimization /36-725
Homework 4 Convex Optimization 10-725/36-725 Due Friday November 4 at 5:30pm submitted to Christoph Dann in Gates 8013 (Remember to a submit separate writeup for each problem, with your name at the top)
More informationConstraint Reduction for Linear Programs with Many Constraints
Constraint Reduction for Linear Programs with Many Constraints André L. Tits Institute for Systems Research and Department of Electrical and Computer Engineering University of Maryland, College Park Pierre-Antoine
More informationLecture 15 Newton Method and Self-Concordance. October 23, 2008
Newton Method and Self-Concordance October 23, 2008 Outline Lecture 15 Self-concordance Notion Self-concordant Functions Operations Preserving Self-concordance Properties of Self-concordant Functions Implications
More informationNonlinear Optimization: What s important?
Nonlinear Optimization: What s important? Julian Hall 10th May 2012 Convexity: convex problems A local minimizer is a global minimizer A solution of f (x) = 0 (stationary point) is a minimizer A global
More informationA Second Full-Newton Step O(n) Infeasible Interior-Point Algorithm for Linear Optimization
A Second Full-Newton Step On Infeasible Interior-Point Algorithm for Linear Optimization H. Mansouri C. Roos August 1, 005 July 1, 005 Department of Electrical Engineering, Mathematics and Computer Science,
More informationConvex Optimization M2
Convex Optimization M2 Lecture 3 A. d Aspremont. Convex Optimization M2. 1/49 Duality A. d Aspremont. Convex Optimization M2. 2/49 DMs DM par email: dm.daspremont@gmail.com A. d Aspremont. Convex Optimization
More informationWritten Examination
Division of Scientific Computing Department of Information Technology Uppsala University Optimization Written Examination 202-2-20 Time: 4:00-9:00 Allowed Tools: Pocket Calculator, one A4 paper with notes
More informationOn Generalized Primal-Dual Interior-Point Methods with Non-uniform Complementarity Perturbations for Quadratic Programming
On Generalized Primal-Dual Interior-Point Methods with Non-uniform Complementarity Perturbations for Quadratic Programming Altuğ Bitlislioğlu and Colin N. Jones Abstract This technical note discusses convergence
More informationSparse Linear Programming via Primal and Dual Augmented Coordinate Descent
Sparse Linear Programg via Primal and Dual Augmented Coordinate Descent Presenter: Joint work with Kai Zhong, Cho-Jui Hsieh, Pradeep Ravikumar and Inderjit Dhillon. Sparse Linear Program Given vectors
More informationTwo hours. To be provided by Examinations Office: Mathematical Formula Tables. THE UNIVERSITY OF MANCHESTER. xx xxxx 2017 xx:xx xx.
Two hours To be provided by Examinations Office: Mathematical Formula Tables. THE UNIVERSITY OF MANCHESTER CONVEX OPTIMIZATION - SOLUTIONS xx xxxx 27 xx:xx xx.xx Answer THREE of the FOUR questions. If
More informationInterior Point Methods for LP
11.1 Interior Point Methods for LP Katta G. Murty, IOE 510, LP, U. Of Michigan, Ann Arbor, Winter 1997. Simplex Method - A Boundary Method: Starting at an extreme point of the feasible set, the simplex
More informationAlgorithms for constrained local optimization
Algorithms for constrained local optimization Fabio Schoen 2008 http://gol.dsi.unifi.it/users/schoen Algorithms for constrained local optimization p. Feasible direction methods Algorithms for constrained
More informationOptimality, Duality, Complementarity for Constrained Optimization
Optimality, Duality, Complementarity for Constrained Optimization Stephen Wright University of Wisconsin-Madison May 2014 Wright (UW-Madison) Optimality, Duality, Complementarity May 2014 1 / 41 Linear
More informationLecture 6: Conic Optimization September 8
IE 598: Big Data Optimization Fall 2016 Lecture 6: Conic Optimization September 8 Lecturer: Niao He Scriber: Juan Xu Overview In this lecture, we finish up our previous discussion on optimality conditions
More informationPrimal-Dual Interior-Point Methods. Ryan Tibshirani Convex Optimization /36-725
Primal-Dual Interior-Point Methods Ryan Tibshirani Convex Optimization 10-725/36-725 Given the problem Last time: barrier method min x subject to f(x) h i (x) 0, i = 1,... m Ax = b where f, h i, i = 1,...
More informationAn O(nL) Infeasible-Interior-Point Algorithm for Linear Programming arxiv: v2 [math.oc] 29 Jun 2015
An O(nL) Infeasible-Interior-Point Algorithm for Linear Programming arxiv:1506.06365v [math.oc] 9 Jun 015 Yuagang Yang and Makoto Yamashita September 8, 018 Abstract In this paper, we propose an arc-search
More informationminimize x x2 2 x 1x 2 x 1 subject to x 1 +2x 2 u 1 x 1 4x 2 u 2, 5x 1 +76x 2 1,
4 Duality 4.1 Numerical perturbation analysis example. Consider the quadratic program with variables x 1, x 2, and parameters u 1, u 2. minimize x 2 1 +2x2 2 x 1x 2 x 1 subject to x 1 +2x 2 u 1 x 1 4x
More informationSecond-Order Cone Program (SOCP) Detection and Transformation Algorithms for Optimization Software
and Second-Order Cone Program () and Algorithms for Optimization Software Jared Erickson JaredErickson2012@u.northwestern.edu Robert 4er@northwestern.edu Northwestern University INFORMS Annual Meeting,
More informationInterior Point Methods. We ll discuss linear programming first, followed by three nonlinear problems. Algorithms for Linear Programming Problems
AMSC 607 / CMSC 764 Advanced Numerical Optimization Fall 2008 UNIT 3: Constrained Optimization PART 4: Introduction to Interior Point Methods Dianne P. O Leary c 2008 Interior Point Methods We ll discuss
More informationAn interior-point gradient method for large-scale totally nonnegative least squares problems
An interior-point gradient method for large-scale totally nonnegative least squares problems Michael Merritt and Yin Zhang Technical Report TR04-08 Department of Computational and Applied Mathematics Rice
More informationOn well definedness of the Central Path
On well definedness of the Central Path L.M.Graña Drummond B. F. Svaiter IMPA-Instituto de Matemática Pura e Aplicada Estrada Dona Castorina 110, Jardim Botânico, Rio de Janeiro-RJ CEP 22460-320 Brasil
More informationAlgorithms for nonlinear programming problems II
Algorithms for nonlinear programming problems II Martin Branda Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Computational Aspects
More informationRachid Benouahboun 1 and Abdelatif Mansouri 1
RAIRO Operations Research RAIRO Oper. Res. 39 25 3 33 DOI:.5/ro:252 AN INTERIOR POINT ALGORITHM FOR CONVEX QUADRATIC PROGRAMMING WITH STRICT EQUILIBRIUM CONSTRAINTS Rachid Benouahboun and Abdelatif Mansouri
More informationConvex Optimization and SVM
Convex Optimization and SVM Problem 0. Cf lecture notes pages 12 to 18. Problem 1. (i) A slab is an intersection of two half spaces, hence convex. (ii) A wedge is an intersection of two half spaces, hence
More informationLecture 1. 1 Conic programming. MA 796S: Convex Optimization and Interior Point Methods October 8, Consider the conic program. min.
MA 796S: Convex Optimization and Interior Point Methods October 8, 2007 Lecture 1 Lecturer: Kartik Sivaramakrishnan Scribe: Kartik Sivaramakrishnan 1 Conic programming Consider the conic program min s.t.
More informationPrimal-Dual Interior-Point Methods. Ryan Tibshirani Convex Optimization
Primal-Dual Interior-Point Methods Ryan Tibshirani Convex Optimization 10-725 Given the problem Last time: barrier method min x subject to f(x) h i (x) 0, i = 1,... m Ax = b where f, h i, i = 1,... m are
More informationOn implementing a primal-dual interior-point method for conic quadratic optimization
On implementing a primal-dual interior-point method for conic quadratic optimization E. D. Andersen, C. Roos, and T. Terlaky December 18, 2000 Abstract Conic quadratic optimization is the problem of minimizing
More informationPrimal-Dual Interior-Point Methods
Primal-Dual Interior-Point Methods Lecturer: Aarti Singh Co-instructor: Pradeep Ravikumar Convex Optimization 10-725/36-725 Outline Today: Primal-dual interior-point method Special case: linear programming
More information5. Duality. Lagrangian
5. Duality Convex Optimization Boyd & Vandenberghe Lagrange dual problem weak and strong duality geometric interpretation optimality conditions perturbation and sensitivity analysis examples generalized
More informationInterior-point methods Optimization Geoff Gordon Ryan Tibshirani
Interior-point methods 10-725 Optimization Geoff Gordon Ryan Tibshirani SVM duality Review min v T v/2 + 1 T s s.t. Av yd + s 1 0 s 0 max 1 T α α T Kα/2 s.t. y T α = 0 0 α 1 Gram matrix K Interpretation
More informationOn Superlinear Convergence of Infeasible Interior-Point Algorithms for Linearly Constrained Convex Programs *
Computational Optimization and Applications, 8, 245 262 (1997) c 1997 Kluwer Academic Publishers. Manufactured in The Netherlands. On Superlinear Convergence of Infeasible Interior-Point Algorithms for
More informationOn Mehrotra-Type Predictor-Corrector Algorithms
On Mehrotra-Type Predictor-Corrector Algorithms M. Salahi, J. Peng, T. Terlaky April 7, 005 Abstract In this paper we discuss the polynomiality of Mehrotra-type predictor-corrector algorithms. We consider
More informationSecond-order cone programming
Outline Second-order cone programming, PhD Lehigh University Department of Industrial and Systems Engineering February 10, 2009 Outline 1 Basic properties Spectral decomposition The cone of squares The
More informationIntroduction to Nonlinear Stochastic Programming
School of Mathematics T H E U N I V E R S I T Y O H F R G E D I N B U Introduction to Nonlinear Stochastic Programming Jacek Gondzio Email: J.Gondzio@ed.ac.uk URL: http://www.maths.ed.ac.uk/~gondzio SPS
More information1.The anisotropic plate model
Pré-Publicações do Departamento de Matemática Universidade de Coimbra Preprint Number 6 OPTIMAL CONTROL OF PIEZOELECTRIC ANISOTROPIC PLATES ISABEL NARRA FIGUEIREDO AND GEORG STADLER ABSTRACT: This paper
More informationLMI Methods in Optimal and Robust Control
LMI Methods in Optimal and Robust Control Matthew M. Peet Arizona State University Lecture 02: Optimization (Convex and Otherwise) What is Optimization? An Optimization Problem has 3 parts. x F f(x) :
More informationIBM Almaden Research Center,650 Harry Road Sun Jose, Calijornia and School of Mathematical Sciences Tel Aviv University Tel Aviv, Israel
and Nimrod Megiddo IBM Almaden Research Center,650 Harry Road Sun Jose, Calijornia 95120-6099 and School of Mathematical Sciences Tel Aviv University Tel Aviv, Israel Submitted by Richard Tapia ABSTRACT
More information1 Outline Part I: Linear Programming (LP) Interior-Point Approach 1. Simplex Approach Comparison Part II: Semidenite Programming (SDP) Concludin
Sensitivity Analysis in LP and SDP Using Interior-Point Methods E. Alper Yldrm School of Operations Research and Industrial Engineering Cornell University Ithaca, NY joint with Michael J. Todd INFORMS
More information15. Conic optimization
L. Vandenberghe EE236C (Spring 216) 15. Conic optimization conic linear program examples modeling duality 15-1 Generalized (conic) inequalities Conic inequality: a constraint x K where K is a convex cone
More informationYinyu Ye, MS&E, Stanford MS&E310 Lecture Note #06. The Simplex Method
The Simplex Method Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye (LY, Chapters 2.3-2.5, 3.1-3.4) 1 Geometry of Linear
More informationAn E cient A ne-scaling Algorithm for Hyperbolic Programming
An E cient A ne-scaling Algorithm for Hyperbolic Programming Jim Renegar joint work with Mutiara Sondjaja 1 Euclidean space A homogeneous polynomial p : E!R is hyperbolic if there is a vector e 2E such
More informationNONLINEAR. (Hillier & Lieberman Introduction to Operations Research, 8 th edition)
NONLINEAR PROGRAMMING (Hillier & Lieberman Introduction to Operations Research, 8 th edition) Nonlinear Programming g Linear programming has a fundamental role in OR. In linear programming all its functions
More informationLecture 5. The Dual Cone and Dual Problem
IE 8534 1 Lecture 5. The Dual Cone and Dual Problem IE 8534 2 For a convex cone K, its dual cone is defined as K = {y x, y 0, x K}. The inner-product can be replaced by x T y if the coordinates of the
More informationComputational Finance
Department of Mathematics at University of California, San Diego Computational Finance Optimization Techniques [Lecture 2] Michael Holst January 9, 2017 Contents 1 Optimization Techniques 3 1.1 Examples
More informationI.3. LMI DUALITY. Didier HENRION EECI Graduate School on Control Supélec - Spring 2010
I.3. LMI DUALITY Didier HENRION henrion@laas.fr EECI Graduate School on Control Supélec - Spring 2010 Primal and dual For primal problem p = inf x g 0 (x) s.t. g i (x) 0 define Lagrangian L(x, z) = g 0
More informationConvex Optimization. Newton s method. ENSAE: Optimisation 1/44
Convex Optimization Newton s method ENSAE: Optimisation 1/44 Unconstrained minimization minimize f(x) f convex, twice continuously differentiable (hence dom f open) we assume optimal value p = inf x f(x)
More informationOn self-concordant barriers for generalized power cones
On self-concordant barriers for generalized power cones Scott Roy Lin Xiao January 30, 2018 Abstract In the study of interior-point methods for nonsymmetric conic optimization and their applications, Nesterov
More informationAlgorithms for nonlinear programming problems II
Algorithms for nonlinear programming problems II Martin Branda Charles University Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Computational Aspects of Optimization
More informationConvex Optimization : Conic Versus Functional Form
Convex Optimization : Conic Versus Functional Form Erling D. Andersen MOSEK ApS, Fruebjergvej 3, Box 16, DK 2100 Copenhagen, Blog: http://erlingdandersen.blogspot.com Linkedin: http://dk.linkedin.com/in/edandersen
More informationOptimisation in Higher Dimensions
CHAPTER 6 Optimisation in Higher Dimensions Beyond optimisation in 1D, we will study two directions. First, the equivalent in nth dimension, x R n such that f(x ) f(x) for all x R n. Second, constrained
More informationA priori bounds on the condition numbers in interior-point methods
A priori bounds on the condition numbers in interior-point methods Florian Jarre, Mathematisches Institut, Heinrich-Heine Universität Düsseldorf, Germany. Abstract Interior-point methods are known to be
More informationA Second-Order Path-Following Algorithm for Unconstrained Convex Optimization
A Second-Order Path-Following Algorithm for Unconstrained Convex Optimization Yinyu Ye Department is Management Science & Engineering and Institute of Computational & Mathematical Engineering Stanford
More informationAnalytic Center Cutting-Plane Method
Analytic Center Cutting-Plane Method S. Boyd, L. Vandenberghe, and J. Skaf April 14, 2011 Contents 1 Analytic center cutting-plane method 2 2 Computing the analytic center 3 3 Pruning constraints 5 4 Lower
More informationLecture: Algorithms for LP, SOCP and SDP
1/53 Lecture: Algorithms for LP, SOCP and SDP Zaiwen Wen Beijing International Center For Mathematical Research Peking University http://bicmr.pku.edu.cn/~wenzw/bigdata2018.html wenzw@pku.edu.cn Acknowledgement:
More informationConic Linear Programming. Yinyu Ye
Conic Linear Programming Yinyu Ye December 2004, revised January 2015 i ii Preface This monograph is developed for MS&E 314, Conic Linear Programming, which I am teaching at Stanford. Information, lecture
More informationLMI MODELLING 4. CONVEX LMI MODELLING. Didier HENRION. LAAS-CNRS Toulouse, FR Czech Tech Univ Prague, CZ. Universidad de Valladolid, SP March 2009
LMI MODELLING 4. CONVEX LMI MODELLING Didier HENRION LAAS-CNRS Toulouse, FR Czech Tech Univ Prague, CZ Universidad de Valladolid, SP March 2009 Minors A minor of a matrix F is the determinant of a submatrix
More informationSemidefinite Programming
Chapter 2 Semidefinite Programming 2.0.1 Semi-definite programming (SDP) Given C M n, A i M n, i = 1, 2,..., m, and b R m, the semi-definite programming problem is to find a matrix X M n for the optimization
More informationThe Q Method for Symmetric Cone Programmin
The Q Method for Symmetric Cone Programming The Q Method for Symmetric Cone Programmin Farid Alizadeh and Yu Xia alizadeh@rutcor.rutgers.edu, xiay@optlab.mcma Large Scale Nonlinear and Semidefinite Progra
More informationSemidefinite Programming
Semidefinite Programming Basics and SOS Fernando Mário de Oliveira Filho Campos do Jordão, 2 November 23 Available at: www.ime.usp.br/~fmario under talks Conic programming V is a real vector space h, i
More informationConstrained Optimization and Lagrangian Duality
CIS 520: Machine Learning Oct 02, 2017 Constrained Optimization and Lagrangian Duality Lecturer: Shivani Agarwal Disclaimer: These notes are designed to be a supplement to the lecture. They may or may
More informationBarrier Method. Javier Peña Convex Optimization /36-725
Barrier Method Javier Peña Convex Optimization 10-725/36-725 1 Last time: Newton s method For root-finding F (x) = 0 x + = x F (x) 1 F (x) For optimization x f(x) x + = x 2 f(x) 1 f(x) Assume f strongly
More informationLagrange Duality. Daniel P. Palomar. Hong Kong University of Science and Technology (HKUST)
Lagrange Duality Daniel P. Palomar Hong Kong University of Science and Technology (HKUST) ELEC5470 - Convex Optimization Fall 2017-18, HKUST, Hong Kong Outline of Lecture Lagrangian Dual function Dual
More informationMore First-Order Optimization Algorithms
More First-Order Optimization Algorithms Yinyu Ye Department of Management Science and Engineering Stanford University Stanford, CA 94305, U.S.A. http://www.stanford.edu/ yyye Chapters 3, 8, 3 The SDM
More information