# A PARAMETRIC SIMPLEX METHOD FOR OPTIMIZING A LINEAR FUNCTION OVER THE EFFICIENT SET OF A BICRITERIA LINEAR PROBLEM. 1.

Save this PDF as:

Size: px
Start display at page:

Download "A PARAMETRIC SIMPLEX METHOD FOR OPTIMIZING A LINEAR FUNCTION OVER THE EFFICIENT SET OF A BICRITERIA LINEAR PROBLEM. 1."

## Transcription

1 ACTA MATHEMATICA VIETNAMICA Volume 21, Number 1, 1996, pp A PARAMETRIC SIMPLEX METHOD FOR OPTIMIZING A LINEAR FUNCTION OVER THE EFFICIENT SET OF A BICRITERIA LINEAR PROBLEM NGUYEN DINH DAN AND LE DUNG MUU Abstract. The problem of optimizing a linear function over the efficient set of a multiple objective problem has many applications in multiple criteria decision making. The main difficulty of this problem is that its feasible region, in general, is a nonconvex set. In this paper we develop a fast algorithm for optimizing a linear function over the efficient set of a bicriteria linear programming problem. The method is a parametric simplex procedure using one parameter in the objective function. 1. Introduction Let X be a nonempty bounded polyhedral convex set in R n given by a system of linear inequalities and/or equalities. Let C denote a (p n)- matrix. The multiple objective linear programming problem (M P ) given by (MP) V max {Cx, x X} is the problem of finding all efficient point of Cx over X. This problem is called bicriteria linear programming problem if C has two rows. We recall that a point x 0 is said to be an efficient point of Cx over X if there no is vector y X such that Cy Cx 0. An efficient point is often called Pareto or nondominated point. Throughout this paper, for two vectors of k-dimensions a = (a 1,..., a k ) and b = (b 1,..., b k ) we write a b (resp. a > b) if and only if a i b i (resp. a i b i, a b) for every i. Let X E denote the efficient set of Problem (MP). The linear optimization problem over the efficient set of (MP) can be stated as Received April 13, 1995; in revised form September 22, Keywords. Optimization over the efficient set, bicriteria, parametric simplex algorithm. This paper is supported in part by the National Basic Research Program in Natural Sciences.

2 60 NGUYEN DINH DAN AND LE DUNG MUU (P) max{d T x, x X E }. Recently, interest in this problem has been intensifying since it has many applications in multiple criteria decision making [11, 12]. Problem (P) can be classified as a global optimization problem since its feasible domain, in general, is a nonconvex set. A few algorithms have been proposed for finding globally optimal solution of (P). Philip [10] first proposed Problem (P) and outlined a cutting plane method for finding an optimal solution of this problem. In [1, 2], using the fact that one can find a simplex S in R p for which Problem (P) is reduced to the infinitely-constrained Problem (Q) formulated as (Q) max d T x subject to λ T Cx λ T Cy y X, x X, λ S, Benson proposed two algorithms for finding a global solution to (P). In both these methods, at each iteration k, Problem (Q) is relaxed by Problem (P k ) given by (P k ) max d T x subject to λ T Cx λ T Cx i (i = 1,..., k) x X, λ S. In the first algorithm this relaxed problem is solved by a branch-and-bound procedure using the convex envelope of the bilinear term λ T Cx. In the second algorithm it is weakened by finding a feasible point (λ k, x k ) such that d T x k > LB k, where LB k is the best known lower bound found at iteration k. By taking h(λ, x) := max y X λ T Cy λ T Cx, Problem (Q) is reduced to the problem max d T x subject to h(λ, x) 0, x X, λ S.

3 A PARAMETRIC SIMPLEX METHOD 61 Since h(λ, x) is a convex-linear function, this problem is a special case of the problem considered in [7]. In [8] an algorithm is proposed which solves (Q) directly. The main computational effort of this algorithm involves searching the vertices of every generated simplex in R p. The algorithm therefore is resonably efficient when p is small while n, the number of the variables, may be fairly large. Perhaps due to its importance and inherent difficulty several algorithmic ideas have also been suggested for solving Problem (P) (see e.g. [3]). In this paper we propose a parametric simplex algorithm for finding a global solution of Problem (P). In the case when X E is the efficient set of a bicriteria linear problem we show that solving (P) amounts to performing a parametric simplex tableau with one parameter in the objective function. In an important special case when d = αc 1 + βc 2 with α 0, β 0, in particular d = c 1 we specialized a result of [4] by showing that a globally optimal solution of Problem (P) can be obtained by solving at most two linear programs. Thus, in this case there exist polynomial time algorithms for solving (P). 2. Preliminaries The algorithm we are going to describe in the next section is based upon the following theorem whose proof can be found, for example in [10]. Theorem 2.1. A vector x 0 is an efficient point of Cx over X if and only if there exists a λ > 0 such that x 0 is an optimal solution for the scalarized problem (L 0 λ ) max{λt Cx, x X}. By dividing to λ i one can always assume that λ i = 1. Thus, in the case p = 2 Problem (L 0 λ ) can be written as (L λ ) max λc 1 + (1 λ)c 2, x, subject to x X, where c 1 and c 2 are the rows of the matrix C. From Theorem 2.1 and the fact that the set of optimal solutions of a linear program is a face of its feasible domain it follows that there exists a finite set I of real numbers such that X E = λ I X λ, where X λ denotes the solution set of the linear program (L 0 λ ). Let ξ(λ) denotes the optimal value of (L 0 λ ). Then solving Problem (P) amounts to solving the following linear programs, one for each λ I, max d T x

4 62 NGUYEN DINH DAN AND LE DUNG MUU subject to (P λ ) x X, λ T Cx = ξ(λ). Let x λ be an optimal solution and η(λ) be the optimal value of this problem. Then η(λ ) := max{η(λ) : λ I} is the globally optimal value and x λ is an optimal solution of Problem (P). 3. A parametric simplex algorithm for linear optimization over the efficient set of a bicriteria linear problem The results described in the previous section suggest applying the parametric simplex method for solving Problem (P). For the case of bicriteria linear problem the algorithm runs as follows. Let λ 1,..., λ K be the critical values of the parametric linear program (L λ ) in the interval [0, 1] (this means that the optimal basic of the linear program (L λ ) is constant in each interval (λ i, λ i+1 ) and changes when λ passes through one λ i ). For each λ i (i = 1,..., K) we solve linear program (L λi ) by the simplex method. From the obtained optimal simplex tableau of (L λi ) we use simplex pivots to obtain the maximal value α i of the linear function d T x over the solution set of (L λi ). Then it is clear that the maximal value of α i (i = 1,..., K) gives the optimal value of the original problem (P). For each λ let c λ := λc 1 + (1 λ)c 2, and denote by α the optimal value of (P). Assume that X := {x R n : ax = b, x 0}, where A is a (m n)-matrix and b R m. Then the algorithm can be described in detail as follows: ALGORITHM Assume that we are given the critical values λ 1,..., λ K of the parametric linear program (L λ ). Let α 0 be a lower bound for α and x 0 X E such that d T x 0 = α 0. Set i = 1. Iteration i (i = 1,..., K) Step 1. Solve the following linear program by the simplex method max < λ i c 1 + (1 λ i )c 2, x >

5 A PARAMETRIC SIMPLEX METHOD 63 subject to x X. Let w i be the obtained optimal basic solution, B 1 = (z jk ) the inverse corresponding basic matrix, and J i be the set of the basic indices. Step 2 (The case when w i is also an optimal solution of (P λi ). If either ik := c λ i k j J i z jk c λ i j < 0 k / J i or d k j J i z jkdj 0 k / J i, then set and α i = d T x i. x i := w i if d T w i > α i 1 and x i = x i 1 otherwise, If i = K, then terminate: x := x i is an optimal solution of Problem (P). If i < K, then increase i by 1 and go to iteration i. Step 3 (The case when w i is not an optimal solution of (P λi )). If d k j j i z jk d j > 0 for some k / J i, then let J + i = {k / J i : c λ i k j J i z jk c λ i j < 0} and solve the linear program (M λi ) given as (M λi ) max{d T x : x X, x k = 0 k J + i }. Let y i be the obtained optimal solution of this linear program, and set x i := y i if d T y i > α i 1 and x i = x i 1 otherwise, and α i = d T x i. Increase i by 1 and go to iteration i. Remarks 1. To solve the linear program (M λi ) it is expedient to start from the obtained solution w i of Program (L λi ). Program (M λi ) can be rewitten in the form max{d T x : x X, λ i c 1 + (1 λ i )c 2 = ξ i }

6 64 NGUYEN DINH DAN AND LE DUNG MUU where ξ i is the optimal value of (L λi ). 2. It is evident that the above algorithm terminates after K iteration yielding a global optimal solution of Problem (P). 3. Sometimes the critical values of the parametric program (L λ ) are not given. In this case, at the beginning of each iteration i we need to compute the critical value λ i by using the parametric simplex method with one parameter in the objective function (see [5]). 4. A Special Case An important special case of Problem (P) occurs when d = c i for some i {1,..., p}. This case have been considered in [1, 4, 7]. For the case when p = 2 and d is a linear combination of the two rows of C, Benson in [4] has shown that the maximal value of d T x over X E attains at a vertex of X which is also an optimal solution of at least one of the following three linear programs: (L i ) max{c i x : x X, } (i = 1, 2) (L) max{c T x : x X, } Using this result Benson [4] give a procedure for maximizing d T x over X E by generating all basic solutions of these linear problems. In this section we specialize this result by observing that an optimal solution of Problem (P ) with d = αc 1 + βc 2 and α 0, β 0 can be obtained by solving at most two linear programs. Namely, we have the following lemma. Lemma 4.1. Let X 2 denote the solution set of the linear program (L 2 ). Then any solution of the program (L 12 ) max{c 1 x : x X 2 } is also an optimal solution of the problem (P 1 ) max{(αc 1 + βc 2 )x : x X E }. Proof. Let x 1 be any solution of (L 12 ). We first show that x 1 is efficient. Indeed, otherwise there exists a point x X such that c i x c i x 1 (i =

7 A PARAMETRIC SIMPLEX METHOD 65 1, 2) and Cx Cx 1. Then, since x 1 X 2, it follows that x X 2 and that c 1 x > c 1 x 1 which contradicts the fact that x 1 is an optimal solution of (L 12 ). Hence x 1 X E. Now let x be a global optimal solution of (P 1 ). Then (αc 1 + βc 2 )x (αc 1 + βc 2 )x 1. From x 1 X 2 it follows that c 2 x 1 c 2 x, and therefore c 1 x 1 c 1 x. This together with x X E implies that c i x 1 = c i x, (i = 1, 2). Hence x 1 is a global optimal solution of (P 1 ). The lemma is proved. By Lemma 4.1, solving Problem (P 1 ) amounts to solving the two linear programs (L 2 ) and (L 12 ). From the linear programming [5] we know that if B is an optimal basic matrix, J B is the set of basic indices and k (k J B ) is the entries of the first row in the simplex tableau corresponding to an optimal solution of (L 2 ), then the solution set of X 2 of Problem (L 2 ), which is the feasible set of Problem (L 12 ), is given by where {x X, x k = 0, k J + B } J + B = {k / J B : k < 0}. Note that if k < 0 for every k / J B, then solving (L 12 ) is avoided, since (L 2 ) has a unique optimal solution. We close the paper by elucidating how to calculate the critical values of the parameters. For simplicity assume that the set X := {x R n : Ax = b, x 0} is nondegenerate and that A is a full rank matrix. Let x be a given optimal solution of the linear program max λc 1 + (1 λc 2, x : x X}. Denote by B the basic matrix corresponding to x, and by J the set of basic indices. If B = (z jk ), then from the linear programming [5] we get k = λc 1 k (1 λ)c 2 k j J z jk (λc 1 j + (1 λ)c 2 j) 0 k / J which implies that (3.1) λ(c 1 k c 2 k (c 1 j c 2 j)) z jk c 2 j c 2 k, k / J. j J j J

8 66 NGUYEN DINH DAN AND LE DUNG MUU (c i k stands for the kth component of the vector ci ). Let m k := c 1 k c 2 k z jk (c 1 j c 2 j), j J t k := j J z jk c 2 j c 2 k, J := {k / J : m k < 0}, J + := {k / J : m k > 0}, α := max{t k /m k : k J }, α + := max{t k /m k : k J + }. Thus, by virtue of (3.1) we have α α +. Moreover, B is also an optimal basic solution of Problem (L λ ) for every λ which belongs to the interval [α, α + ]. References 1. H. P. Benson, An All-Linear Programming Relaxation Algorithm for Optimizing Over the Efficient Set, J. of Global Optimization 1 (1991), , A Finite, Nonadjacent Extreme-Point Search Algorithm for Optimization Over the Efficient Set, J. of Optimization Theory and Applications 73 (1992), H. P. Benson and S. Sayin, A Face Search Heuristic Algorithm for Optimizing Over the Efficient Set, Naval Research Logistics 40 (1993), , Optimization over Over the Efficient Set: Four Special cases, J. of Optimization Theory and Applications, to appear. 5. H. Danzig, Linear Programming and Its Extension, Princeton Press R. Horst and H. Tuy, Global Optimization (Deterministic approaches), Spriger- Verlag, Berlin H. Isermann and R. E. Steuer, Computational Experience Concerning Payoff Tables and Minimum Criterion Values Over the Efficient Set, European J. of Operational Research 33 (1987), Le D. Muu, An Algorithm for Solving Convex Programs with and Additional Convex-Concave Constraints, Mathematical Programming 61 (1993), Le D. Muu, A Method for Optimization of a Linear Function over the Efficient Set, J. of Global Optimization, to appear. 10. J. Philip, Algorithms for the Vector Maximization Problem, Mathematical Programming 2 (1972), R. E. Steuer, Multiple Criteria Optimization Theory: Computation and Application, John Wiley, New York P. L. Yu, Multiple criteria Decision Making, New York 1985.

9 A PARAMETRIC SIMPLEX METHOD 67 Polytechnical University of Hanoi Institute of Mathematics, Box 631, Hanoi

### Bilinear Programming

Bilinear Programming Artyom G. Nahapetyan Center for Applied Optimization Industrial and Systems Engineering Department University of Florida Gainesville, Florida 32611-6595 Email address: artyom@ufl.edu

### The Simplex Algorithm

The Simplex Algorithm Uri Feige November 2011 1 The simplex algorithm The simplex algorithm was designed by Danzig in 1947. This write-up presents the main ideas involved. It is a slight update (mostly

### Graphs that have the feasible bases of a given linear

Algorithmic Operations Research Vol.1 (2006) 46 51 Simplex Adjacency Graphs in Linear Optimization Gerard Sierksma and Gert A. Tijssen University of Groningen, Faculty of Economics, P.O. Box 800, 9700

### OPERATIONS RESEARCH. Linear Programming Problem

OPERATIONS RESEARCH Chapter 1 Linear Programming Problem Prof. Bibhas C. Giri Department of Mathematics Jadavpur University Kolkata, India Email: bcgiri.jumath@gmail.com 1.0 Introduction Linear programming

### MA4254: Discrete Optimization. Defeng Sun. Department of Mathematics National University of Singapore Office: S Telephone:

MA4254: Discrete Optimization Defeng Sun Department of Mathematics National University of Singapore Office: S14-04-25 Telephone: 6516 3343 Aims/Objectives: Discrete optimization deals with problems of

### / Approximation Algorithms Lecturer: Michael Dinitz Topic: Linear Programming Date: 2/24/15 Scribe: Runze Tang

600.469 / 600.669 Approximation Algorithms Lecturer: Michael Dinitz Topic: Linear Programming Date: 2/24/15 Scribe: Runze Tang 9.1 Linear Programming Suppose we are trying to approximate a minimization

### In this chapter we introduce some of the basic concepts that will be useful for the study of integer programming problems.

2 Basics In this chapter we introduce some of the basic concepts that will be useful for the study of integer programming problems. 2.1 Notation Let A R m n be a matrix with row index set M = {1,...,m}

### DC Programming: A brief tutorial. Andres Munoz Medina Courant Institute of Mathematical Sciences

DC Programming: A brief tutorial. Andres Munoz Medina Courant Institute of Mathematical Sciences munoz@cims.nyu.edu Difference of Convex Functions Definition: A function exists f is said to be DC if there

### Topological properties of convex sets

Division of the Humanities and Social Sciences Ec 181 KC Border Convex Analysis and Economic Theory Winter 2018 Topic 5: Topological properties of convex sets 5.1 Interior and closure of convex sets Let

### Discrete Optimization. Lecture Notes 2

Discrete Optimization. Lecture Notes 2 Disjunctive Constraints Defining variables and formulating linear constraints can be straightforward or more sophisticated, depending on the problem structure. The

### Compact Sets. James K. Peterson. September 15, Department of Biological Sciences and Department of Mathematical Sciences Clemson University

Compact Sets James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 15, 2017 Outline 1 Closed Sets 2 Compactness 3 Homework Closed Sets

### Numerical Optimization

Convex Sets Computer Science and Automation Indian Institute of Science Bangalore 560 012, India. NPTEL Course on Let x 1, x 2 R n, x 1 x 2. Line and line segment Line passing through x 1 and x 2 : {y

### Lesson 17. Geometry and Algebra of Corner Points

SA305 Linear Programming Spring 2016 Asst. Prof. Nelson Uhan 0 Warm up Lesson 17. Geometry and Algebra of Corner Points Example 1. Consider the system of equations 3 + 7x 3 = 17 + 5 = 1 2 + 11x 3 = 24

### VARIANTS OF THE SIMPLEX METHOD

C H A P T E R 6 VARIANTS OF THE SIMPLEX METHOD By a variant of the Simplex Method (in this chapter) we mean an algorithm consisting of a sequence of pivot steps in the primal system using alternative rules

### Division of the Humanities and Social Sciences. Convex Analysis and Economic Theory Winter Separation theorems

Division of the Humanities and Social Sciences Ec 181 KC Border Convex Analysis and Economic Theory Winter 2018 Topic 8: Separation theorems 8.1 Hyperplanes and half spaces Recall that a hyperplane in

### 16.410/413 Principles of Autonomy and Decision Making

16.410/413 Principles of Autonomy and Decision Making Lecture 17: The Simplex Method Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology November 10, 2010 Frazzoli (MIT)

### SUBSTITUTING GOMORY CUTTING PLANE METHOD TOWARDS BALAS ALGORITHM FOR SOLVING BINARY LINEAR PROGRAMMING

Bulletin of Mathematics Vol. 06, No. 0 (20), pp.. SUBSTITUTING GOMORY CUTTING PLANE METHOD TOWARDS BALAS ALGORITHM FOR SOLVING BINARY LINEAR PROGRAMMING Eddy Roflin, Sisca Octarina, Putra B. J Bangun,

### 3 No-Wait Job Shops with Variable Processing Times

3 No-Wait Job Shops with Variable Processing Times In this chapter we assume that, on top of the classical no-wait job shop setting, we are given a set of processing times for each operation. We may select

### An FPTAS for minimizing the product of two non-negative linear cost functions

Math. Program., Ser. A DOI 10.1007/s10107-009-0287-4 SHORT COMMUNICATION An FPTAS for minimizing the product of two non-negative linear cost functions Vineet Goyal Latife Genc-Kaya R. Ravi Received: 2

### FACES OF CONVEX SETS

FACES OF CONVEX SETS VERA ROSHCHINA Abstract. We remind the basic definitions of faces of convex sets and their basic properties. For more details see the classic references [1, 2] and [4] for polytopes.

### Approximation Algorithms: The Primal-Dual Method. My T. Thai

Approximation Algorithms: The Primal-Dual Method My T. Thai 1 Overview of the Primal-Dual Method Consider the following primal program, called P: min st n c j x j j=1 n a ij x j b i j=1 x j 0 Then the

### A modification of the simplex method reducing roundoff errors

A modification of the simplex method reducing roundoff errors Ralf Ulrich Kern Georg-Simon-Ohm-Hochschule Nürnberg Fakultät Informatik Postfach 21 03 20 90121 Nürnberg Ralf-Ulrich.Kern@ohm-hochschule.de

### Lecture 15: The subspace topology, Closed sets

Lecture 15: The subspace topology, Closed sets 1 The Subspace Topology Definition 1.1. Let (X, T) be a topological space with topology T. subset of X, the collection If Y is a T Y = {Y U U T} is a topology

### On vertex-coloring edge-weighting of graphs

Front. Math. China DOI 10.1007/s11464-009-0014-8 On vertex-coloring edge-weighting of graphs Hongliang LU 1, Xu YANG 1, Qinglin YU 1,2 1 Center for Combinatorics, Key Laboratory of Pure Mathematics and

### arxiv: v1 [math.co] 27 Feb 2015

Mode Poset Probability Polytopes Guido Montúfar 1 and Johannes Rauh 2 arxiv:1503.00572v1 [math.co] 27 Feb 2015 1 Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103 Leipzig, Germany,

### Convexity: an introduction

Convexity: an introduction Geir Dahl CMA, Dept. of Mathematics and Dept. of Informatics University of Oslo 1 / 74 1. Introduction 1. Introduction what is convexity where does it arise main concepts and

### Mathematical Programming and Research Methods (Part II)

Mathematical Programming and Research Methods (Part II) 4. Convexity and Optimization Massimiliano Pontil (based on previous lecture by Andreas Argyriou) 1 Today s Plan Convex sets and functions Types

### Optimization of Design. Lecturer:Dung-An Wang Lecture 8

Optimization of Design Lecturer:Dung-An Wang Lecture 8 Lecture outline Reading: Ch8 of text Today s lecture 2 8.1 LINEAR FUNCTIONS Cost Function Constraints 3 8.2 The standard LP problem Only equality

### Introductory Operations Research

Introductory Operations Research Theory and Applications Bearbeitet von Harvir Singh Kasana, Krishna Dev Kumar 1. Auflage 2004. Buch. XI, 581 S. Hardcover ISBN 978 3 540 40138 4 Format (B x L): 15,5 x

### Investigating Mixed-Integer Hulls using a MIP-Solver

Investigating Mixed-Integer Hulls using a MIP-Solver Matthias Walter Otto-von-Guericke Universität Magdeburg Joint work with Volker Kaibel (OvGU) Aussois Combinatorial Optimization Workshop 2015 Outline

### EULER S FORMULA AND THE FIVE COLOR THEOREM

EULER S FORMULA AND THE FIVE COLOR THEOREM MIN JAE SONG Abstract. In this paper, we will define the necessary concepts to formulate map coloring problems. Then, we will prove Euler s formula and apply

### Rigidity, connectivity and graph decompositions

First Prev Next Last Rigidity, connectivity and graph decompositions Brigitte Servatius Herman Servatius Worcester Polytechnic Institute Page 1 of 100 First Prev Next Last Page 2 of 100 We say that a framework

### BCN Decision and Risk Analysis. Syed M. Ahmed, Ph.D.

Linear Programming Module Outline Introduction The Linear Programming Model Examples of Linear Programming Problems Developing Linear Programming Models Graphical Solution to LP Problems The Simplex Method

Iterative Improvement Algorithm design technique for solving optimization problems Start with a feasible solution Repeat the following step until no improvement can be found: change the current feasible

### 5 The Theory of the Simplex Method

5 The Theory of the Simplex Method Chapter 4 introduced the basic mechanics of the simplex method. Now we shall delve a little more deeply into this algorithm by examining some of its underlying theory.

### Generalized Convex Set-Valued Maps

Generalized Convex Set-Valued Maps September 20, 2008 Joël Benoist Department of Mathematics LACO URA-CNRS 1586 University of Limoges 87060 Limoges, France E-mail: benoist@unilim.fr Nicolae Popovici Faculty

### The complement of PATH is in NL

340 The complement of PATH is in NL Let c be the number of nodes in graph G that are reachable from s We assume that c is provided as an input to M Given G, s, t, and c the machine M operates as follows:

### DEGENERACY AND THE FUNDAMENTAL THEOREM

DEGENERACY AND THE FUNDAMENTAL THEOREM The Standard Simplex Method in Matrix Notation: we start with the standard form of the linear program in matrix notation: (SLP) m n we assume (SLP) is feasible, and

### SUBSTITUTING GOMORY CUTTING PLANE METHOD TOWARDS BALAS ALGORITHM FOR SOLVING BINARY LINEAR PROGRAMMING

ASIAN JOURNAL OF MATHEMATICS AND APPLICATIONS Volume 2014, Article ID ama0156, 11 pages ISSN 2307-7743 http://scienceasia.asia SUBSTITUTING GOMORY CUTTING PLANE METHOD TOWARDS BALAS ALGORITHM FOR SOLVING

### REINER HORST. License or copyright restrictions may apply to redistribution; see

MATHEMATICS OF COMPUTATION Volume 66, Number 218, April 1997, Pages 691 698 S 0025-5718(97)00809-0 ON GENERALIZED BISECTION OF n SIMPLICES REINER HORST Abstract. A generalized procedure of bisection of

### Convexity. 1 X i is convex. = b is a hyperplane in R n, and is denoted H(p, b) i.e.,

Convexity We ll assume throughout, without always saying so, that we re in the finite-dimensional Euclidean vector space R n, although sometimes, for statements that hold in any vector space, we ll say

### Solution of m 3 or 3 n Rectangular Interval Games using Graphical Method

Australian Journal of Basic and Applied Sciences, 5(): 1-10, 2011 ISSN 1991-8178 Solution of m or n Rectangular Interval Games using Graphical Method Pradeep, M. and Renukadevi, S. Research Scholar in

### Principles and Practical Applications of the Optimal Values Management and Programming

Principles and Practical Applications of the Optimal Values Management and Programming Nderim Zeqiri State University of Tetova, Faculty of Applied Sciences, Tetovo, Macedonia Received May 05, 2016; Accepted

### Open and Closed Sets

Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.

### Method and Algorithm for solving the Bicriterion Network Problem

Proceedings of the 00 International Conference on Industrial Engineering and Operations Management Dhaka, Bangladesh, anuary 9 0, 00 Method and Algorithm for solving the Bicriterion Network Problem Hossain

### 4.1 Graphical solution of a linear program and standard form

4.1 Graphical solution of a linear program and standard form Consider the problem min c T x Ax b x where x = ( x1 x ) ( 16, c = 5 ), b = 4 5 9, A = 1 7 1 5 1. Solve the problem graphically and determine

### COMS 4771 Support Vector Machines. Nakul Verma

COMS 4771 Support Vector Machines Nakul Verma Last time Decision boundaries for classification Linear decision boundary (linear classification) The Perceptron algorithm Mistake bound for the perceptron

### Polynomial-Time Approximation Algorithms

6.854 Advanced Algorithms Lecture 20: 10/27/2006 Lecturer: David Karger Scribes: Matt Doherty, John Nham, Sergiy Sidenko, David Schultz Polynomial-Time Approximation Algorithms NP-hard problems are a vast

### This lecture: Convex optimization Convex sets Convex functions Convex optimization problems Why convex optimization? Why so early in the course?

Lec4 Page 1 Lec4p1, ORF363/COS323 This lecture: Convex optimization Convex sets Convex functions Convex optimization problems Why convex optimization? Why so early in the course? Instructor: Amir Ali Ahmadi

### arxiv: v1 [math.co] 12 Dec 2017

arxiv:1712.04381v1 [math.co] 12 Dec 2017 Semi-reflexive polytopes Tiago Royer Abstract The Ehrhart function L P(t) of a polytope P is usually defined only for integer dilation arguments t. By allowing

### A Comparative study on Algorithms for Shortest-Route Problem and Some Extensions

International Journal of Basic & Applied Sciences IJBAS-IJENS Vol: No: 0 A Comparative study on Algorithms for Shortest-Route Problem and Some Extensions Sohana Jahan, Md. Sazib Hasan Abstract-- The shortest-route

### Applied Lagrange Duality for Constrained Optimization

Applied Lagrange Duality for Constrained Optimization Robert M. Freund February 10, 2004 c 2004 Massachusetts Institute of Technology. 1 1 Overview The Practical Importance of Duality Review of Convexity

### The Branch-and-Sandwich Algorithm for Mixed-Integer Nonlinear Bilevel Problems

The Branch-and-Sandwich Algorithm for Mixed-Integer Nonlinear Bilevel Problems Polyxeni-M. Kleniati and Claire S. Adjiman MINLP Workshop June 2, 204, CMU Funding Bodies EPSRC & LEVERHULME TRUST OUTLINE

### MATH 890 HOMEWORK 2 DAVID MEREDITH

MATH 890 HOMEWORK 2 DAVID MEREDITH (1) Suppose P and Q are polyhedra. Then P Q is a polyhedron. Moreover if P and Q are polytopes then P Q is a polytope. The facets of P Q are either F Q where F is a facet

### Optimality certificates for convex minimization and Helly numbers

Optimality certificates for convex minimization and Helly numbers Amitabh Basu Michele Conforti Gérard Cornuéjols Robert Weismantel Stefan Weltge October 20, 2016 Abstract We consider the problem of minimizing

### On the null space of a Colin de Verdière matrix

On the null space of a Colin de Verdière matrix László Lovász 1 and Alexander Schrijver 2 Dedicated to the memory of François Jaeger Abstract. Let G = (V, E) be a 3-connected planar graph, with V = {1,...,

### PRIMAL-DUAL INTERIOR POINT METHOD FOR LINEAR PROGRAMMING. 1. Introduction

PRIMAL-DUAL INTERIOR POINT METHOD FOR LINEAR PROGRAMMING KELLER VANDEBOGERT AND CHARLES LANNING 1. Introduction Interior point methods are, put simply, a technique of optimization where, given a problem

### AN ALGORITHM FOR A MINIMUM COVER OF A GRAPH

AN ALGORITHM FOR A MINIMUM COVER OF A GRAPH ROBERT Z. NORMAN1 AND MICHAEL O. RABIN2 Introduction. This paper contains two similar theorems giving conditions for a minimum cover and a maximum matching of

### 60 2 Convex sets. {x a T x b} {x ã T x b}

60 2 Convex sets Exercises Definition of convexity 21 Let C R n be a convex set, with x 1,, x k C, and let θ 1,, θ k R satisfy θ i 0, θ 1 + + θ k = 1 Show that θ 1x 1 + + θ k x k C (The definition of convexity

### ROUGH MEMBERSHIP FUNCTIONS: A TOOL FOR REASONING WITH UNCERTAINTY

ALGEBRAIC METHODS IN LOGIC AND IN COMPUTER SCIENCE BANACH CENTER PUBLICATIONS, VOLUME 28 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1993 ROUGH MEMBERSHIP FUNCTIONS: A TOOL FOR REASONING

### Discrete Optimization 2010 Lecture 5 Min-Cost Flows & Total Unimodularity

Discrete Optimization 2010 Lecture 5 Min-Cost Flows & Total Unimodularity Marc Uetz University of Twente m.uetz@utwente.nl Lecture 5: sheet 1 / 26 Marc Uetz Discrete Optimization Outline 1 Min-Cost Flows

### GENERAL ASSIGNMENT PROBLEM via Branch and Price JOHN AND LEI

GENERAL ASSIGNMENT PROBLEM via Branch and Price JOHN AND LEI Outline Review the column generation in Generalized Assignment Problem (GAP) GAP Examples in Branch and Price 2 Assignment Problem The assignment

### Section Notes 5. Review of Linear Programming. Applied Math / Engineering Sciences 121. Week of October 15, 2017

Section Notes 5 Review of Linear Programming Applied Math / Engineering Sciences 121 Week of October 15, 2017 The following list of topics is an overview of the material that was covered in the lectures

### Solution for Homework set 3

TTIC 300 and CMSC 37000 Algorithms Winter 07 Solution for Homework set 3 Question (0 points) We are given a directed graph G = (V, E), with two special vertices s and t, and non-negative integral capacities

### Orientation of manifolds - definition*

Bulletin of the Manifold Atlas - definition (2013) Orientation of manifolds - definition* MATTHIAS KRECK 1. Zero dimensional manifolds For zero dimensional manifolds an orientation is a map from the manifold

### OPTIMAL LINK CAPACITY ASSIGNMENTS IN TELEPROCESSING AND CENTRALIZED COMPUTER NETWORKS *

OPTIMAL LINK CAPACITY ASSIGNMENTS IN TELEPROCESSING AND CENTRALIZED COMPUTER NETWORKS * IZHAK RUBIN UCLA Los Angeles, California Summary. We consider a centralized network model representing a teleprocessing

### FUZZY METRIC SPACES ZUN-QUAN XIA AND FANG-FANG GUO

J. Appl. Math. & Computing Vol. 16(2004), No. 1-2, pp. 371-381 FUZZY METRIC SPACES ZUN-QUAN XIA AND FANG-FANG GUO Abstract. In this paper, fuzzy metric spaces are redefined, different from the previous

### Steiner Trees and Forests

Massachusetts Institute of Technology Lecturer: Adriana Lopez 18.434: Seminar in Theoretical Computer Science March 7, 2006 Steiner Trees and Forests 1 Steiner Tree Problem Given an undirected graph G

### EE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 6

EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 6 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory April 19, 2012 Andre Tkacenko

### On Rainbow Cycles in Edge Colored Complete Graphs. S. Akbari, O. Etesami, H. Mahini, M. Mahmoody. Abstract

On Rainbow Cycles in Edge Colored Complete Graphs S. Akbari, O. Etesami, H. Mahini, M. Mahmoody Abstract In this paper we consider optimal edge colored complete graphs. We show that in any optimal edge

### Modules. 6 Hamilton Graphs (4-8 lectures) Introduction Necessary conditions and sufficient conditions Exercises...

Modules 6 Hamilton Graphs (4-8 lectures) 135 6.1 Introduction................................ 136 6.2 Necessary conditions and sufficient conditions............. 137 Exercises..................................

### LECTURE 13: SOLUTION METHODS FOR CONSTRAINED OPTIMIZATION. 1. Primal approach 2. Penalty and barrier methods 3. Dual approach 4. Primal-dual approach

LECTURE 13: SOLUTION METHODS FOR CONSTRAINED OPTIMIZATION 1. Primal approach 2. Penalty and barrier methods 3. Dual approach 4. Primal-dual approach Basic approaches I. Primal Approach - Feasible Direction

### Two Characterizations of Hypercubes

Two Characterizations of Hypercubes Juhani Nieminen, Matti Peltola and Pasi Ruotsalainen Department of Mathematics, University of Oulu University of Oulu, Faculty of Technology, Mathematics Division, P.O.

### Improved Results on Geometric Hitting Set Problems

Improved Results on Geometric Hitting Set Problems Nabil H. Mustafa nabil@lums.edu.pk Saurabh Ray saurabh@cs.uni-sb.de Abstract We consider the problem of computing minimum geometric hitting sets in which,

### Le D. Muu and Werner Oettli

An Algorithm for ndefinite Quadratk with Convex Constraints Programming Le D. Muu and Werner Oettli Nr. 89 (1989) Fakultät für Mathematik und nformatik, Universität Mannheim, D-6800 Mannheim 1, Germany

### LECTURE 7 LECTURE OUTLINE. Review of hyperplane separation Nonvertical hyperplanes Convex conjugate functions Conjugacy theorem Examples

LECTURE 7 LECTURE OUTLINE Review of hyperplane separation Nonvertical hyperplanes Convex conjugate functions Conjugacy theorem Examples Reading: Section 1.5, 1.6 All figures are courtesy of Athena Scientific,

### The Geometry of Carpentry and Joinery

The Geometry of Carpentry and Joinery Pat Morin and Jason Morrison School of Computer Science, Carleton University, 115 Colonel By Drive Ottawa, Ontario, CANADA K1S 5B6 Abstract In this paper we propose

### Convex sets and convex functions

Convex sets and convex functions Convex optimization problems Convex sets and their examples Separating and supporting hyperplanes Projections on convex sets Convex functions, conjugate functions ECE 602,

### Ramsey s Theorem on Graphs

Ramsey s Theorem on Graphs 1 Introduction Exposition by William Gasarch Imagine that you have 6 people at a party. We assume that, for every pair of them, either THEY KNOW EACH OTHER or NEITHER OF THEM

### The Encoding Complexity of Network Coding

The Encoding Complexity of Network Coding Michael Langberg Alexander Sprintson Jehoshua Bruck California Institute of Technology Email: mikel,spalex,bruck @caltech.edu Abstract In the multicast network

### IE 5531: Engineering Optimization I

IE 5531: Engineering Optimization I Lecture 3: Linear Programming, Continued Prof. John Gunnar Carlsson September 15, 2010 Prof. John Gunnar Carlsson IE 5531: Engineering Optimization I September 15, 2010

### Lecture 11 COVERING SPACES

Lecture 11 COVERING SPACES A covering space (or covering) is not a space, but a mapping of spaces (usually manifolds) which, locally, is a homeomorphism, but globally may be quite complicated. The simplest

### 1.7 The Heine-Borel Covering Theorem; open sets, compact sets

1.7 The Heine-Borel Covering Theorem; open sets, compact sets This section gives another application of the interval halving method, this time to a particularly famous theorem of analysis, the Heine Borel

### AN APPROXIMATION APPROACH FOR RANKING FUZZY NUMBERS BASED ON WEIGHTED INTERVAL - VALUE 1.INTRODUCTION

Mathematical and Computational Applications, Vol. 16, No. 3, pp. 588-597, 2011. Association for Scientific Research AN APPROXIMATION APPROACH FOR RANKING FUZZY NUMBERS BASED ON WEIGHTED INTERVAL - VALUE

### Rank Bounds and Integrality Gaps for Cutting Planes Procedures

" THEORY OF COMPUTING, Volume 0 (2005), pp. 78 100 http://theoryofcomputing.org Rank Bounds and Integrality Gaps for Cutting Planes Procedures Joshua Buresh-Oppenheim Nicola Galesi Shlomo Hoory Avner Magen

### Minimum sum multicoloring on the edges of trees

Minimum sum multicoloring on the edges of trees Dániel Marx a,1 a Department of Computer Science and Information Theory, Budapest University of Technology and Economics, H-1521 Budapest, Hungary. Abstract

### A Note on the Succinctness of Descriptions of Deterministic Languages

INFORMATION AND CONTROL 32, 139-145 (1976) A Note on the Succinctness of Descriptions of Deterministic Languages LESLIE G. VALIANT Centre for Computer Studies, University of Leeds, Leeds, United Kingdom

### COMPOSITION OF STABLE SET POLYHEDRA

COMPOSITION OF STABLE SET POLYHEDRA Benjamin McClosky and Illya V. Hicks Department of Comptational and Applied Mathematics Rice University November 30, 2007 Abstract Barahona and Mahjob fond a defining

### Optimization Methods: Advanced Topics in Optimization - Multi-objective Optimization 1. Module 8 Lecture Notes 2. Multi-objective Optimization

Optimization Methods: Advanced Topics in Optimization - Multi-objective Optimization 1 Module 8 Lecture Notes 2 Multi-objective Optimization Introduction In a real world problem it is very unlikely that

### Control point based representation of inellipses of triangles

Annales Mathematicae et Informaticae 40 (2012) pp. 37 46 http://ami.ektf.hu Control point based representation of inellipses of triangles Imre Juhász Department of Descriptive Geometry, University of Miskolc,

### On Truck dock assignment problem with operational time constraint within cross docks

On Truck dock assignment problem with operational time constraint within cross docks Shahin Gelareh a,b,, Gilles Goncalves a,b, Rahimeh N. Monemi c,1, a UArtois, LGI2A, F-62400, Béthune, France b Univ

### Alternating Projections

Alternating Projections Stephen Boyd and Jon Dattorro EE392o, Stanford University Autumn, 2003 1 Alternating projection algorithm Alternating projections is a very simple algorithm for computing a point

### Topology and Topological Spaces

Topology and Topological Spaces Mathematical spaces such as vector spaces, normed vector spaces (Banach spaces), and metric spaces are generalizations of ideas that are familiar in R or in R n. For example,

### 2. CONNECTIVITY Connectivity

2. CONNECTIVITY 70 2. Connectivity 2.1. Connectivity. Definition 2.1.1. (1) A path in a graph G = (V, E) is a sequence of vertices v 0, v 1, v 2,..., v n such that {v i 1, v i } is an edge of G for i =

### A Simplex Based Parametric Programming Method for the Large Linear Programming Problem

A Simplex Based Parametric Programming Method for the Large Linear Programming Problem Huang, Rujun, Lou, Xinyuan Abstract We present a methodology of parametric objective function coefficient programming

### On the Complexity of the Policy Improvement Algorithm. for Markov Decision Processes

On the Complexity of the Policy Improvement Algorithm for Markov Decision Processes Mary Melekopoglou Anne Condon Computer Sciences Department University of Wisconsin - Madison 0 West Dayton Street Madison,

### On some subclasses of circular-arc graphs

On some subclasses of circular-arc graphs Guillermo Durán - Min Chih Lin Departamento de Computación Facultad de Ciencias Exactas y Naturales Universidad de Buenos Aires e-mail: {willy,oscarlin}@dc.uba.ar

### = w. w u. u ; u + w. x x. z z. y y. v + w. . Remark. The formula stated above is very important in the theory of. surface integral.

1 Chain rules 2 Directional derivative 3 Gradient Vector Field 4 Most Rapid Increase 5 Implicit Function Theorem, Implicit Differentiation 6 Lagrange Multiplier 7 Second Derivative Test Theorem Suppose