DOCSLIB.ORG

Generalization of the Primal and Dual Affine Scaling Algorithms

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Generalization of the Primal and Dual Affine Scaling Algorithms Generalization of the primal and dual a±ne scaling algorithms F. G. M. CUNHAy, A. W. M. PINTOz, P. R. OLIVEIRA¤x and J. X. da C. NETO{ yDepartment of Mathematics, Cear¶aFederal Centre of Technological Education, Fortaleza, Brazil; zDepartment of Mathematics, Federal University of Amazonas, Manaus, Brazil; xDepartment of Systems and Computing Engineering, Rio de Janeiro Federal University, Rio de Janeiro, Brazil; {Department of Mathematics, Federal University of Piau¶³,Teresina, Brazil; (v1.0 ¯nished 23 April 2005) We obtain a new class of primal a±ne scaling algorithms for the linearly constrained convex program- ming. It is constructed through a family of metrics generated by ¡r power, r ¸ 1, of the diagonal iterate vector matrix. We prove the so-called weak convergence. It generalizes some known algo- rithms. Working in dual space, we generalize the dual a±ne scaling algorithm of Adler, Karmarkar, Resende and Veiga, similarly depending on a r-parameter and we give its global convergence proof for nondegenerate linear programs. With the purpose of observing the computational performance of the methods, we compare them with classical algorithms (when r = 1 or r = 2), implementing the proposed families and applying to some linear programs obtained from NETLIB library. In the case of primal family, we also apply it to some quadratic programming problems described in the Maros and M¶esz¶arosrepository. Keywords: Interior Point Algorithms; A±ne Scaling Algorithms; Linear Convex Programming 1 Introduction We are interested in a±ne scaling potential algorithms. If we restrict ourselves to primal interior point methods which is one class we are interested in, we mention the a±ne scaling algorithm for linear and quadratic cost functions [1] and the polynomial logarithmic barrier approach for convex self-concordant functions, as can be seen in [2]. As our algorithm can be easily particularized to the case without linear constraints, i.e., the minimization over the positive orthant, we mention the multiplicative method [3], analysed in [4]. ¤Corresponding author. Tel.: +55 21 2562-8675; Fax: +55 21 2562-8676; E-mail: [email protected] yE-mail: [email protected]; zE-mail: [email protected]; {E-mail: [email protected] 2 F. G. M. Cunha et al. n In our approach, we let the positive orthant R++ as a Riemannian manifold. The knowledge of Riemannian geometry concepts is not necessary to under- stand this paper, but those interested can see: [5] for Riemannian geometry and [6,7] for that regarding speci¯c connection with optimization. The connec- tion between geometry and mathematical programming comes from [8]. Since then, some researches have been carried out (see, e.g. [6,7,9{16]). Most of the papers consider the so-called geodesic algorithms, where the usual straight line search of the cost function is substituted by the minimization along the geodesics which has, in general, a high computing cost. Our method follows the usual pattern | one line search is performed along the given direction. Following the same lines as this paper, we have proposed classes of algorithms for the nonlinear complementarity problem in [17] and for the minimization over the positive orthant within a proximal method set in [18]. The directions of our class, corresponding to r = 1 and r = 2 are, respec- tively, the classic multiplicative and a±ne scaling directions. The convergence result we got is the so-called weak-convergence, as obtained in [4] for the mul- tiplicative method. Finally, let us mention the power a±ne scaling method [19] for linear programming, whose convergence results are also valid for the de- generate case. Working in dual space we propose for linear programs, a generalization of the dual a±ne scaling algorithm of Adler, Karmarkar, Resende and Veiga [20], similarly depending on a r-parameter. This generalization is based on metrics de¯ned on dual space. We give the global convergence proof of the algorithm for nondegenerate problems. The paper is organized as follows. In section 2, we present the generalized primal a±ne scaling algorithm giving some background theory and its conver- gence. In section 3, we present the generalized dual a±ne scaling algorithm and give its global convergence proof for nondegenerate linear programs. Section 4 is dedicated to observing the computational performance of the methods and comparing them with classical algorithms (when r = 1 or r = 2). The paper ends reaching some conclusions and suggesting possible future researches. 2 Generalized Primal A±ne Scaling Method 2.1 Introduction We consider the resolution of (P) minimizex f(x) subject to Ax = b (1) x ¸ 0; Generalization of the primal and dual a±ne scaling algorithms 3 where the objective function f : Rn ! R is a di®erentiable and convex func- tion, A is a m £ n matrix and b a Rm vector. n Associated with R++, we de¯ne a metric through the diagonal matrix ¡r G(x) = X , for r ¸ 1, X being the diagonal© matrix whose nonnullª ele- n n ments are the x¡ 2 R++ entries.¢ We take M = x 2 R++ : Ax = b as a n ¡r submanifold of R++; X , r ¸ 1, with the induced metric. As we show below, considering f de¯ned on M, the opposite of its gradient is the a±ne scaling direction corresponding to each r ¸ 1. We limit ourselves to presenting the theoretical results we need to construct the algorithm direction. Those interested in the detailed description of the concepts presented here can see [7]. We will now obtain the gradient of f restricted to M. The ¯rst element we need is the tangent space to M, the usual null space of A, given by n TxM = TM = fd 2 R : Ad = 0g ; (2) for any x (it would be x dependent for nonlinear constraints). In order, we need the expression of the projection operator onto TM. De- noting I as the identity n £ n matrix, the projection onto TM, under any (positive de¯nite matrix) metric G, is ¡1 T ¡ ¡1 T ¢¡1 PG(x) = I ¡ G (x)A AG (x)A A: (3) In our case, with G = X¡r, this rewrites ¡ ¢¡1 P (x) = I ¡ XrAT AXrAT A: (4) ¡ ¢ n ¡r Now, notice that the gradient of f restricted to R++; X is ¡1 r r n f(x) = G rf(x) = X rf(x); (5) R++ n for any x 2 R++, where r represents the usual (Euclidean) gradient. Therefore, the gradient of f restricted to (M; X¡r) is the projection of r n f(x) onto TM, i.e., R++ r rM f(x) = P (x)X rf(x): It is easy to check that rM f(x) 2 TMp. For all d 2 TM such that kdkG := hd; Gdi = 1, using Cauchy{Schwartz 4 F. G. M. Cunha et al. inequality, we obtain jhd; rM f(x)iGj · kdkGkrM f(x)kG = krM f(x)kG: Consequently, ¡rM f(x)=krM f(x)kG is the steepest descent direction of f at x. Since f is convex, x¤ is a solution for (1) if and only if there exist Lagrangian ¤ m ¤ n multipliers y 2 R and s 2 R+ such that AT y¤ + s¤ = rf(x¤) Ax¤ = b ¤ ¤ (6) xi si = 0; i = 1; :::; n x¤ ¸ 0: From (6), some simple calculations lead to ¡ ¢ ¤ r T ¡1 r ¤ y(x ) = AX¤A AX¤rf(x ) s(x¤) = rf(x¤) ¡ AT y(x¤) (7) r ¤ X¤s = 0: Therefore, it is natural to de¯ne the following functions on M: ¡ ¢¡1 y(x) ´ AXrAT AXrrf(x); (8) s(x) ´ rf(x) ¡ AT y(x) = P T (x)rf(x); (9) r r d(x) ´ X s(x) = P (x)X rf(x) = rM f(x): (10) Under the nondegeneracy assumption, all above functions can be extended to the closure of M. Now, we apply the gradient direction computed to de¯ne the generalized primal a±ne scaling algorithm (GPAS). Algorithm GPAS: initialization: Let (A; b; c; x0 2 M; stopping criterion; ¯ > 0 and ± 2 (0; 1)) k = 0 Generalization of the primal and dual a±ne scaling algorithms 5 while stopping criterion not satis¯ed ¡ ¢ k r T ¡1 r k y = AXkA AXkrf(x ) sk = rf(xk) ¡ AT yk k r k d = X¡ks ¢ k ¡1 k ¹ =  Xk d ; 0 k ± ® = ¯+¹k k k k t = arg mint2[0;®k] f(x ¡ td ) xk+1 = xk ¡ tkdk k = k + 1 end while end Algorithm GPAS where Â(v) denotes the largest component of v. An immediate consequence is the fact that the algorithm is an interior point method. Indeed, for xk 2 M, k ¸ 0, it holds that ³ ´ ¡1 k+1 k k ¡1 k k Xk x = 1 ¡ t (xi ) di ¸ 1; if di · 0; i otherwise, we have ±¹k tk(xk)¡1dk · < 1: i i ¯ + ¹k k k ¡1 k 0 Therefore, 1¡t (xi ) di > 0. Since x > 0, the assertion follows by induction. To assure a large search interval [0; ®k], the safety factors ± must be near 1, and ¯ near 0. 2.2 Convergence of the Generalized Primal A±ne Scaling Algorithm In this subsection we will study the convergence of the generalized primal a±ne scaling algorithm with the following assumptions: Assumption 1: The matrix A has full rank m; Assumption 2: The primal problem (P) has an interior feasible solution, i.e., P ++ = fx 2 Rn : Ax = b; x > 0g 6= ;; Assumption 3: (P) is a nondegenerate problem;© ª 0 Assumption 4: The starting level set, Lx0 = x 2 M : f(x) · f(x ) , is 6 F. G. M. Cunha et al.
Load more

Recommended publications
  • Using Ant System Optimization Technique for Approximate Solution to Multi- Objective Programming Problems
    International Journal of Scientific & Engineering Research, Volume 4, Issue 9, September-2013 ISSN 2229-5518 1701 Using Ant System Optimization Technique for Approximate Solution to Multi- objective Programming Problems M. El-sayed Wahed and Amal A. Abou-Elkayer Department of Computer Science, Faculty of Computers and Information Suez Canal University (EGYPT) E-mail: [email protected], Abstract In this paper we use the ant system optimization metaheuristic to find approximate solution to the Multi-objective Linear Programming Problems (MLPP), the advantageous and disadvantageous of the suggested method also discussed focusing on the parallel computation and real time optimization, it's worth to mention here that the suggested method doesn't require any artificial variables the slack and surplus variables are enough, a test example is given at the end to show how the method works. Keywords: Multi-objective Linear. Programming. problems, Ant System Optimization Problems Introduction The Multi-objective Linear. Programming. problems is an optimization method which can be used to find solutions to problems where the Multi-objective function and constraints are linear functions of the decision variables, the intersection of constraints result in a polyhedronIJSER which represents the region that contain all feasible solutions, the constraints equation in the MLPP may be in the form of equalities or inequalities, and the inequalities can be changed to equalities by using slack or surplus variables, one referred to Rao [1], Philips[2], for good start and introduction to the MLPP. The LP type of optimization problems were first recognized in the 30's by economists while developing methods for the optimal allocations of resources, the main progress came from George B.
    [Show full text]
  • An Annotated Bibliography of Network Interior Point Methods
    AN ANNOTATED BIBLIOGRAPHY OF NETWORK INTERIOR POINT METHODS MAURICIO G.C. RESENDE AND GERALDO VEIGA Abstract. This paper presents an annotated bibliography on interior point methods for solving network flow problems. We consider single and multi- commodity network flow problems, as well as preconditioners used in imple- mentations of conjugate gradient methods for solving the normal systems of equations that arise in interior network flow algorithms. Applications in elec- trical engineering and miscellaneous papers complete the bibliography. The collection includes papers published in journals, books, Ph.D. dissertations, and unpublished technical reports. 1. Introduction Many problems arising in transportation, communications, and manufacturing can be modeled as network flow problems (Ahuja et al. (1993)). In these problems, one seeks an optimal way to move flow, such as overnight mail, information, buses, or electrical currents, on a network, such as a postal network, a computer network, a transportation grid, or a power grid. Among these optimization problems, many are special classes of linear programming problems, with combinatorial properties that enable development of efficient solution techniques. The focus of this annotated bibliography is on recent computational approaches, based on interior point methods, for solving large scale network flow problems. In the last two decades, many other approaches have been developed. A history of computational approaches up to 1977 is summarized in Bradley et al. (1977). Several computational studies established the fact that specialized network simplex algorithms are orders of magnitude faster than the best linear programming codes of that time. (See, e.g. the studies in Glover and Klingman (1981); Grigoriadis (1986); Kennington and Helgason (1980) and Mulvey (1978)).
    [Show full text]
  • An Affine-Scaling Pivot Algorithm for Linear Programming
    AN AFFINE-SCALING PIVOT ALGORITHM FOR LINEAR PROGRAMMING PING-QI PAN¤ Abstract. We proposed a pivot algorithm using the affine-scaling technique. It produces a sequence of interior points as well as a sequence of vertices, until reaching an optimal vertex. We report preliminary but favorable computational results obtained with a dense implementation of it on a set of standard Netlib test problems. Key words. pivot, interior point, affine-scaling, orthogonal factorization AMS subject classifications. 65K05, 90C05 1. Introduction. Consider the linear programming (LP) problem in the stan- dard form min cT x (1.1) subject to Ax = b; x ¸ 0; where c 2 Rn; b 2 Rm , A 2 Rm£n(m < n). Assume that rank(A) = m and c 62 range(AT ). So the trivial case is eliminated where the objective value cT x is constant over the feasible region of (1.1). The assumption rank(A) = m is not substantial either to the proposed algorithm,and will be dropped later. As a LP problem solver, the simplex algorithm might be one of the most famous and widely used mathematical tools in the world. Its philosophy is to move on the underlying polyhedron, from a vertex to adjacent vertex, along edges until an optimal vertex is reached. Since it was founded by G.B.Dantzig [8] in 1947, the simplex algorithm has been very successful in practice, and occupied a dominate position in this area, despite its infiniteness in case of degeneracy. However, it turned out that the simplex algorithm may require an exponential amount of time to solve LP problems in the worst-case [29].
    [Show full text]
  • On Some Polynomial-Time Algorithms for Solving Linear Programming Problems
    Available online a t www.pelagiaresearchlibrary.com Pelagia Research Library Advances in Applied Science Research, 2012, 3 (5):3367-3373 ISSN: 0976-8610 CODEN (USA): AASRFC On Some Polynomial-time Algorithms for Solving Linear Programming Problems B. O. Adejo and H. S. Adaji Department of Mathematical Sciences, Kogi State University, Anyigba _____________________________________________________________________________________________ ABSTRACT In this article we survey some Polynomial-time Algorithms for Solving Linear Programming Problems namely: the ellipsoid method, Karmarkar’s algorithm and the affine scaling algorithm. Finally, we considered a test problem which we solved with the methods where applicable and conclusions drawn from the results so obtained. __________________________________________________________________________________________ INTRODUCTION An algorithm A is said to be a polynomial-time algorithm for a problem P, if the number of steps (i. e., iterations) required to solve P on applying A is bounded by a polynomial function O(m n,, L) of dimension and input length of the problem. The Ellipsoid method is a specific algorithm developed by soviet mathematicians: Shor [5], Yudin and Nemirovskii [7]. Khachian [4] adapted the ellipsoid method to derive the first polynomial-time algorithm for linear programming. Although the algorithm is theoretically better than the simplex algorithm, which has an exponential running time in the worst case, it is very slow practically and not competitive with the simplex method. Nevertheless, it is a very important theoretical tool for developing polynomial-time algorithms for a large class of convex optimization problems. Narendra K. Karmarkar is an Indian mathematician; renowned for developing the Karmarkar’s algorithm. He is listed as an ISI highly cited researcher.
    [Show full text]
  • An Affine-Scaling Interior-Point Method for Continuous Knapsack Constraints with Application to Support Vector Machines∗
    SIAM J. OPTIM. c 2011 Society for Industrial and Applied Mathematics Vol. 21, No. 1, pp. 361–390 AN AFFINE-SCALING INTERIOR-POINT METHOD FOR CONTINUOUS KNAPSACK CONSTRAINTS WITH APPLICATION TO SUPPORT VECTOR MACHINES∗ MARIA D. GONZALEZ-LIMA†, WILLIAM W. HAGER‡ , AND HONGCHAO ZHANG§ Abstract. An affine-scaling algorithm (ASL) for optimization problems with a single linear equality constraint and box restrictions is developed. The algorithm has the property that each iterate lies in the relative interior of the feasible set. The search direction is obtained by approximating the Hessian of the objective function in Newton’s method by a multiple of the identity matrix. The algorithm is particularly well suited for optimization problems where the Hessian of the objective function is a large, dense, and possibly ill-conditioned matrix. Global convergence to a stationary point is established for a nonmonotone line search. When the objective function is strongly convex, ASL converges R-linearly to the global optimum provided the constraint multiplier is unique and a nondegeneracy condition holds. A specific implementation of the algorithm is developed in which the Hessian approximation is given by the cyclic Barzilai-Borwein (CBB) formula. The algorithm is evaluated numerically using support vector machine test problems. Key words. interior-point, affine-scaling, cyclic Barzilai-Borwein methods, global convergence, linear convergence, support vector machines AMS subject classifications. 90C06, 90C26, 65Y20 DOI. 10.1137/090766255 1. Introduction. In this paper we develop an interior-point algorithm for a box-constrained optimization problem with a linear equality constraint: (1.1) min f(x) subject to x ∈ Rn, l ≤ x ≤ u, aTx = b.
    [Show full text]
  • Interior Point Method
    LECTURE 6: INTERIOR POINT METHOD 1. Motivation 2. Basic concepts 3. Primal affine scaling algorithm 4. Dual affine scaling algorithm Motivation • Simplex method works well in general, but suffers from exponential-time computational complexity. • Klee-Minty example shows simplex method may have to visit every vertex to reach the optimal one. • Total complexity of an iterative algorithm = # of iterations x # of operations in each iteration • Simplex method - Simple operations: Only check adjacent extreme points - May take many iterations: Klee-Minty example Question: any fix? • Complexity of the simplex method Worst case performance of the simplex method Klee-Minty Example: •Victor Klee, George J. Minty, “How good is the simplex algorithm?’’ in (O. Shisha edited) Inequalities, Vol. III (1972), pp. 159-175. Klee-Minty Example Klee-Minty Example Karmarkar’s (interior point) approach • Basic idea: approach optimal solutions from the interior of the feasible domain • Take more complicated operations in each iteration to find a better moving direction • Require much fewer iterations General scheme of an interior point method • An iterative method that moves in the interior of the feasible domain Interior movement (iteration) • Given a current interior feasible solution , we have > 0 An interior movement has a general format Key knowledge • 1. Who is in the interior? - Initial solution • 2. How do we know a current solution is optimal? - Optimality condition • 3. How to move to a new solution? - Which direction to move? (good feasible direction) - How far to go? (step-length) Q1 - Who is in the interior? • Standard for LP • Who is at the vertex? • Who is on the edge? • Who is on the boundary? • Who is in the interior? What have learned before Who is in the interior? • Two criteria for a point x to be an interior feasible solution: 1.
    [Show full text]
  • Generalized Affine Scaling Algorithms for Linear Programming Problems
    Generalized Affine Scaling Algorithms for Linear Programming Problems Md Sarowar Morshed and Md Noor-E-Alam Department of Mechanical & Industrial Engineering Northeastern University 360 Huntington Avenue, Boston, MA 02115, USA Email : [email protected] Abstract Interior Point Methods are widely used to solve Linear Programming problems. In this work, we present two primal Affine Scaling algorithms to achieve faster convergence in solving Linear Programming problems. In the first algorithm, we integrate Nesterov's restarting strategy in the primal Affine Scaling method with an extra parameter, which in turn generalizes the original primal Affine Scaling method. We provide the proof of convergence for the proposed generalized al- gorithm considering long step size. We also provide the proof of convergence for the primal and dual sequence without the degeneracy assumption. This con- vergence result generalizes the original convergence result for the Affine Scaling methods and it gives us hints about the existence of a new family of meth- ods. Then, we introduce a second algorithm to accelerate the convergence rate of the generalized algorithm by integrating a non-linear series transformation technique. Our numerical results show that the proposed algorithms outperform the original primal Affine Scaling method. Key words: Linear Programming, Affine Scaling, Nesterov Acceleration, Dikin Process, Shanks Series Transformation 1 Introduction The Affine Scaling (AFS) algorithm was introduced by Dikin [1], which re- mained unnoticed to the Operations Research (OR) community until the seminal work of Karmarkar [2]. Karmarkar's work transformed the research in Interior Point Methods (IPMs) and induced a significant development in the theory of arXiv:1804.07373v4 [math.OC] 12 Oct 2019 IPMs.
    [Show full text]
  • Affine Scaling I Orithll1s
    7 Affine Scaling I orithll1s Since its introduction in 1984, Karmarkar's projective scaling algorithm has become the most notable interior-point method for solving linear programming problems. This pioneering work has stimulated a flurry of research activities in the field. Among all reported variants of Karmarkar' s original algorithm, the affine scaling approach especially attracted researchers' attention. This approach uses the simple affine transformation to replace Karmarkar's original projective transformation and allows people to work on the linear programming problems in standard form. The special simplex structure required by Karmarkar' s algorithm is relaxed. The basic affine scaling algorithm was first presented by I. I. Dikin, a Soviet mathematician, in 1967. Later, in 1985, the work was independently rediscovered by E. Barnes and R. Vanderbei, M. Meketon, and B. Freedman. They proposed using the (primal) affine scaling algorithm to solve the (primal) linear programs in standard form and established convergence proof of the algorithm. A similar algorithm, the so-called dual affine scaling algorithm, was designed and implemented by I. Adler, N. Karmarkar, M. G. C. Resende, and G. Veiga for solving (dual) linear programs in inequality form. Compared to the relatively cumbersome projective transformation, the implementation of both the primal and dual affine scaling algorithms become quite straightforward. These two algorithms are currently the variants subject to the widest experimentation and exhibit promising results, although the theoretical proof of polynomial-time complexity was lost in the simplified transformation. In fact, N. Megiddo and M. Shub's work indicated that the trajectory leading to the optimal solution provided by the basic affine scaling algorithms depends upon the starting solution.
    [Show full text]
  • Interior-Point Algorithms for Linear-Programming Decoding
    Interior-Point Algorithms for Linear-Programming Decoding0 Pascal O. Vontobel Hewlett–Packard Laboratories Palo Alto, CA 94304, USA Email: [email protected] Abstract— Interior-point algorithms constitute a very interest- for which it can easily be shown that all codewords in C are ing class of algorithms for solving linear-programming problems. vertices of P. In this paper we study efficient implementations of such algo- The same polytope P appeared also in papers by Koetter rithms for solving the linear program that appears in the linear- programming decoder formulation. and Vontobel [3], [4], [5], where message-passing iterative (MPI) decoders were analyzed and where this polytope P was I. INTRODUCTION called the fundamental polytope. The appearance of the same Consider a binary linear code C of length n that is used object in these two different contexts suggests that there is a for data transmission over a binary-input discrete memoryless tight connection between LP decoding and MPI decoding. channel. As was observed by Feldman et al. [1], [2], the ML The above codes Cj can be any codes of length n, however, decoder for this setup can be written as in the following we will focus on the case where these codes are codes of dimension n−1. For example, let H be an m×n T ˆxML = arg min hγ, xi, parity-check matrix for the code C and let h be the j-th row x∈C j of H.1 Then, defining where γ is a length-n vector that contains the log-likelihood x n h x ratios and where hγ, xi is the inner product (in R) of the Cj = ∈{0, 1} h j , i =0 (mod 2) vector γ with the vector x.
    [Show full text]
  • A New Class of Scaling Matrices for Scaled Trust Region Algorithms
    ISSN: 2377-3316 http://dx.doi.org/10.21174/josetap.v3i1. 44 A New Class of Scaling Matrices for Scaled Trust Region Algorithms Aydin Ayanzadeh*1, Shokoufeh Yazdanian2, Ehsan Shahamatnia3 1 Department of Applied Informatics, Informatics Institute, Istanbul Technical University, Istanbul, Turkey 2Department of Computer science, University of Tabriz, Tabriz, Iran 3Department of Computer and Electrical Engineering, Universidade Nova de Lisboa, Lisbon, Portugal Abstract— A new class of affine scaling matrices for the interior point Newton-type methods is considered to solve the nonlinear systems with simple bounds. We review the essential properties of a scaling matrix and consider several well- known scaling matrices proposed in the literature. We define a new scaling matrix that is the convex combination of these matrices. The proposed scaling matrix inherits those interesting properties of the individual matrices and satisfies additional desired requirements. The numerical experiments demonstrate the superiority of the new scaling matrix in solving several important test problems. Keywords— Diagonal Scaling; Bound-Constrained Equations; Scaling Matrices; Trust Region Methods; Affine Scaling. I. INTRODUCTION where FX: n is a continuously differentiable Diverse applications including signal processing mapping, X n is an open set containing the n- [36] and compressive sensing [3,4] incorporate many dimensional box ={x n | l x u }. The vectors convex nonlinear optimization problems. Although nn are lower and upper specialized algorithms have been developed for some of lu( { − }) , ( { + }) these problems, the interior-point methods are still the bounds on the variables such that has a nonempty main tool to tackle them. These methods require a interior. feasible initial point [19].
    [Show full text]
  • Affine-Scaling Method, 202–204, 206, 210 Poor Performance Of
    Index affine-scaling method, 202–204, 206, 210 blocking variable, 48 poor performance of, 203 breakpoint, 163 steplength choice, 203 approximation problems, 218–227 canonical form, 8, 16, 45, 117, 120, 123, 1-norm, 221–224 129, 130, 134, 138, 139, 150, 2-norm. See least-squares problems 151, 156, 195, 197 ∞-norm. See Chebyshev approximation dual of, 109 for linear equalities, 218–223 for problems with bound constraints, with hard constraints, 223–224, 227 130 arc, 12–14, 144 centering parameter, 206, 214 central path, 205 back substitution. See triangular substitution Chebyshev approximation, 219–221, 223 basic feasible solution, 118 Cholesky factorization, 209, 212 as iterates of revised simplex method, classification, 11–12, 230–235 123 labeled points, 230 association with basis matrix, 120–121 testing set, 235 association with vertex, 121–123 training set, 235 degenerate, 120 tuning set, 235 existence, 119–120 classifier, 230 initial, 129, 134–139 linear, 230 basic solution, 118 nonlinear, 233, 234 basis, 117, 126, 152 complementarity, 101–102, 178, 237 for problems with bound constraints, almost-, 178, 179 130 existence of strictly complementary initial, 125, 134–137 primal-dual solution, 114–115 optimal, 129, 152–156 strict, 101 basis matrix, 52, 118, 120, 123, 139, 140, complementary slackness. 154 See complementarity for network linear program, 149 concave function, 161, 244 LU factorization of, 139–142 constraints, 1 big M method, 110–112 active, 14 behavior on infeasible problems, 112 bounds, 74–75 motivation, 110–111 equality, 72–87 proof of effectiveness, 111 inactive, 14 specification, 111–112 inequality, 72–87 bimatrix game, 185 convex combination, 239, 241 Bland’s rule.
    [Show full text]
  • Interior Point Methods of Mathematical Programming Applied Optimization
    Interior Point Methods of Mathematical Programming Applied Optimization Volume 5 Series Editors: Panos M. Pardalos University of Florida, U.SA. Donald Hearn University ofFlorida, U.S.A. The titles published in this series are listed at the end of this volume. Interior Point Methods of Mathematical Programming Edited by Tamas Terlaky Delft University a/Technology KLUWER ACADEMIC PUBLISHERS DORDRECHT I BOSTON I LONDON A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN-13: 978-1-4613-3451-4 e-ISBN-13: 978-1-4613-3449-1 DOl: 10.1007/978-1-4613-3449-1 Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus NiJ'hoff, Dr W. Junk and MTP Press. Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper All Rights Reserved @ 19% Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. This book is dedicated to the memory of Professor Gyorgy Sonnevend, the father of analytic centers. CONTENTS PREFACE xv Part I LINEAR PROGRAMMING
    [Show full text]