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UCSD - MATH - 171A

Math 140A, Fall 2010, Midterm, 11/8/10Instructions. Answer all questions. You may use without proof anything which was proved in class. Cite a theorem either by name, if it has one, or by briefly stating what it says. 1. (20 points) Give an example of an

UCSD - MATH - 171A

Math 140A, Fall 2010, Quiz, 10/15/10Instructions. Answer all questions. You may use without proof anything which was proved in class. If you need to cite a theorem, do so either by name, if it has one, or by briefly stating what it says. 1. (10 points) L

UCSD - MATH - 171A

Math 171A: Linear ProgrammingOverview of Math 171ALecture 1 Overview of the Class: Introduction to OptimizationPhilip E. Gillc 2011Class text:P. E. Gill, W. Murray and M. H. Wright, Numerical Linear Algebra and Optimization, Addison-Wesley Publishin

UCSD - MATH - 171A

Math 171A LINEAR PROGRAMMING Class Notesc 1998. Philip E. Gill, Walter Murray and Margaret H. Wright Department of Mathematics University of California, San Diego, La Jolla, CA 92093-0112. January 2007Contents1 Background 1.1. Denitions and Operations

UCSD - MATH - 171A

Math 171A: Linear ProgrammingOverview of Math 171ALecture 1 Overview of the Class: Introduction to OptimizationPhilip E. Gillc 2011Class text:P. E. Gill, W. Murray and M. H. Wright, Numerical Linear Algebra and Optimization, Addison-Wesley Publishin

UCSD - MATH - 171A

RecapMath 171A: Linear ProgrammingLecture 2 Properties of Linear ConstraintsPhilip E. Gillc 2011The lecture slides and homework are posted on the class web-page. http:/ccom.ucsd.edu/~peg/math171a Access to course materials requires a class account an

UCSD - MATH - 171A

Recap: a linear inequality constraintx2Math 171A: Linear ProgrammingLecture 3 Geometry of the Feasible RegionPhilip E. Gillc 2011aT x > b aT x = b aT x < bhttp:/ccom.ucsd.edu/~peg/math171aFriday, January 7th, 2011x1UCSD Center for Computational

UCSD - MATH - 171A

Recap: Properties of linear constraintsMath 171A: Linear Programmingconstraint #1: constraint #2: constraint #3: constraint #4: constraint #5: constraint #6:The constraints may be infeasibleLecture 4 Properties of the Objective FunctionPhilip E. Gill

UCSD - MATH - 171A

Recap: basic properties of an LPMath 171A: Linear ProgrammingLecture 5 Review of Linear Equations IPhilip E. Gillc 2011An LP is either infeasible, unbounded or has an optimal solution. An optimal solution always lies on the boundary of the feasible r

UCSD - MATH - 171A

Recap: Two fundamental subspacesMath 171A: Linear Programmingrange(A)= =cfw_y : y = Axfor some x Rn Lecture 6 Full-Rank Systems of Linear EquationsPhilip E. Gillc 2011range(AT ) The set range(A)cfw_x : x = AT y for some y Rm "lives" in Rm , i.e

UCSD - MATH - 171A

Math 171A: Linear ProgrammingSo far, we have focused on compatible equations Ax = b. i.e., equations Ax = b with b range(A).Lecture 7 Properties of Incompatible SystemsQuestionPhilip E. Gillc 2011Given A Rmn , how do we characterize vectors b Rm suc

UCSD - MATH - 171A

Math 171A: Linear ProgrammingClass AnnouncementsLecture 8 Linear Programming with Equality ConstraintsPhilip E. Gillc 20111The midterm will be held in class next Wednesday, January 26. The midterm is based on material covered in Homework Assignments

UCSD - MATH - 171A

Math 171A: Linear ProgrammingRecap: LP with equality constraintsLecture 9 Optimality Conditions for LP with Equality ConstraintsPhilip E. Gillc 2011Linear programming with equality constraints: ELP minimize nxRc Tx Ax = bsubject to http:/ccom.ucsd

UCSD - MATH - 171A

Recap: Optimality conditions for ELPMath 171A: Linear ProgrammingLinear programming with equality constraints: ELP minimize nxRLecture 10 Feasible Directions and VerticesPhilip E. Gillc 2011c Tx Ax = bsubject to A point x is optimal if and only if

UCSD - MATH - 171A

Recap: computing the step to a constraintMath 171A: Linear ProgrammingLecture 11 VerticesPhilip E. Gillc 2011http:/ccom.ucsd.edu/~peg/math171aThe step to the constraint aiTx bi from a feasible point x along a nonzero p is: ri (x) if aiTp = 0 -aTp i

UCSD - MATH - 171A

Recap: Vertices (aka corner points)Math 171A: Linear ProgrammingLecture 12 Finding a VertexPhilip E. Gillc 2011A vertex is a feasible point at which there are at least n linearly independent constraints active. i.e., the active-constraint matrix Aa h

UCSD - MATH - 171A

Recap: convex setsMath 171A: Linear Programming Definition (Convex set)Lecture 13 Optimality Conditions for LPA set S is convex if, for every x, y S, it holds that z = (1 - )x + y S Note that for all 0 1Philip E. Gillc 2011z = (1 - )x + y = x + (y -

UCSD - MATH - 171A

Recap: Optimality conditions for LPMath 171A: Linear ProgrammingLecture 14 Farkas' Lemma and its ImplicationsPhilip E. Gillc 2011LPminimizexc Tx Ax bsubject towith A an m n matrix, b an m-vector and c an n-vector. The vector x is a solution of L

UCSD - MATH - 171A

Recap: Farkas' lemmaMath 171A: Linear ProgrammingLecture 15 Testing for OptimalityPhilip E. Gillc 2011c Tp 0 for all p such that Aa p 0 if and only if c = AT for some 0 a a ahttp:/ccom.ucsd.edu/~peg/math171aWednesday, February 9th, 2011UCSD Center

UCSD - MATH - 171A

Math 171A: Linear ProgrammingThe simplex method defines a sequence of vertices:Lecture 16 The Simplex Methodsuch that Philip E. Gillc 2011x0 , x1 , x2 , . . . , x k , . . . , vertex numberc Tx0 c Tx1 c Tx2 c Txk For the moment, we assume that a sta

UCSD - MATH - 171A

Math 171A: Linear ProgrammingRecap: the simplex methodLecture 17 Degenerate Constraints and the Simplex MethodPhilip E. Gillc 2011One step moves from a vertex to an adjacent vertex. Each step requires the solution of two sets of equations: ATk = c k

UCSD - MATH - 171A

Recap: the simplex methodMath 171A: Linear ProgrammingOne step moves from a vertex to an adjacent vertex.Lecture 18 Finding a Feasible PointPhilip E. Gillc 2011Each step requires the solution of two sets of equations: ATk = c k and Ak pk = eswhere

UCSD - MATH - 171A

Recap: The phase-1 Linear ProgramMath 171A: Linear ProgrammingTo find a feasible x for minimize nxRc Tx Ax b , x 0simple boundsLecture 19 Finding a Feasible Point IIsubject togeneral constraintsSolve the linear program: Philip E. Gillc 2011mini

UCSD - MATH - 171A

Recap: LP formulationsMath 171A: Linear ProgrammingProblems considered so far: ELP minimize nxRLecture 20 LPs with Mixed Constraint TypesPhilip E. Gillc 2011c Tx Ax = b c Tx Ax bsubject to LP minimizexhttp:/ccom.ucsd.edu/~peg/math171asubject to

UCSD - MATH - 171A

Recap: Linear programs in standard formMath 171A: Linear ProgrammingLecture 21 Linear Programs in Standard FormPhilip E. Gillc 2011minimize nxRc Tx Ax = b , x 0simple boundssubject toequality constraintsThe matrix A is m n with shape A = We app

UCSD - MATH - 171A

Recap: Simplex method for LPs in standard formMath 171A: Linear ProgrammingLecture 22 The Simplex Method for Standard FormPhilip E. Gillc 2011minimize nxRc Tx Ax = b , x 0simple boundssubject toequality constraintsWe apply "mixed-constraint" si

UCSD - MATH - 171A

Recap: Choice of the generic formMath 171A: Linear Programmingminimize d Tw nw Rsubject to Gw f ,w 0Lecture 23 Linear Programming DualityConvert to all-inequality form if m > n, i.e.,G= Philip E. Gillc 2011http:/ccom.ucsd.edu/~peg/math171aConve

UCSD - MATH - 171A

Recap: DualityMath 171A: Linear ProgrammingProblem conversion involves defining a primal linear program and converting it into another dual linear program Primal in all-inequality form: (P): minimize nxRLecture 24 Linear Programming Duality IIPhilip

UCSD - MATH - 171A

Math 171A: Linear ProgrammingLecture 25 Complexity of the Simplex MethodPhilip E. Gillc 2011Given a particular instance of a linear program, can we predict how long it will take to solve it? There are two issues:1 2how much time does it take to perf

UCSD - MATH - 171A

Math 171A: Linear ProgrammingConsider an LP in standard form: minimize nxRc Tx x 0Lecture 26 Interior Methods for Linear ProgrammingPhilip E. Gillc 2011subject to Ax = b, where A has m rows. The optimality conditions are: AT + z = c, Ax = b, x z =

UCSD - MATH - 171A

Math 171A Homework 1 SolutionsInstructor: Jiawang Nie January 24, 20121. Find the optimal solution to the following LP maximize x1 + 4x2 + 3x3 subject to 2x1 + x2 + x3 4 x1 - x3 = 1 x2 0, x3 0. Solution: The second equality constraint x1 = x3 + 1 allows

UCSD - MATH - 171A

Math 171A Homework 2 SolutionsInstructor: Jiawang Nie February 1, 20121. (8 points) Consider the feasible set F defined by the following constraints x1 + x2 4, x1 + 3x2 6, 6x1 - x2 18, 3 x2 6, x1 -1.(a) Express F in the standard form Ax b. Write down A

UCSD - MATH - 171A

Math 171A Homework 3 SolutionsInstructor: Jiawang Nie February 1, 20121. (4 points) For a given nonzero matrix A and nonzero vector b, assume that b may be written as b = bR + bN , where bR Range(A) and bN Null(AT ). (a) Show that bR and bN are unique.

UCSD - MATH - 171A

Math 171A Homework 4 SolutionsInstructor: Jiawang Nie February 14, 20121. (4 points) Consider the matrix of active constraints of an LP at a certain point x -1 -8 4 -7 -3 -4 2 0 4 -2 . Aa () = x -6 0 -3 3 5 0 0 5 2 7 Use Matlab to find a direction p alo

UCSD - MATH - 171A

Math 171A Homework 5 SolutionsInstructor: Jiawang Nie February 21, 20121. (5 points) Consider the linear program Minimize 3x1 subject to -2x1 x1 -2x1 2x1 x1 x2 -x2 +2x3 +4x2 +4x3 +4x2 +x3 +x2 +2x3 -2x2 -3x2 +x3 6 5 1 0 -2 0 0 x3 0and the point x = (1,

UCSD - MATH - 171A

Math 171A Homework 6 SolutionsInstructor: Jiawang Nie March 4, 20121. (5 points) Consider the LP of minimizing cT x subject to Ax b where 1 0 1 A= 1 0 0 1 1 0 0 1 0 0 1 1 , 0 0 1 2 5 4 b = , c = AT 1 1 1 0 1 1 . 1 0 0Find the minimizer of this LP by us

UCSD - MATH - 171A

Math 171A Practice Midterm IIInstructor: Jiawang Nie March 5, 20121. Consider the LP of minimizing c 2 3 A = 4 2 3x subject to Ax b where -15 3 5 -13 4 3 , b = -20 , c = A 5 1 -12 4 4 -13 5 2 0 1 0 . 1 1Find the minimizer of this LP by using optimali

UCSD - MATH - 171A

Math 171A Homework 1 SolutionsInstructor: Jiawang Nie January 24, 20121. Find the optimal solution to the following LP maximize x1 + 4x2 + 3x3 subject to 2x1 + x2 + x3 4 x1 - x3 = 1 x2 0, x3 0. Solution: The second equality constraint x1 = x3 + 1 allows

UCSD - MATH - 171A

Math 171A: Linear ProgrammingInstructor: Philip E. GillWinter Quarter 2011Homework Assignment #1 Due Friday January 14, 2011 I know that you are all aware of the importance of doing the homework assignments. This is the best way to keep up with the cla

UCSD - MATH - 171A

Math 171A: Linear ProgrammingInstructor: Philip E. GillWinter Quarter 2011Homework Assignment #1 Due Friday January 14, 2011 I know that you are all aware of the importance of doing the homework assignments. This is the best way to keep up with the cla

UCSD - MATH - 171A

Math 171A: Linear ProgrammingInstructor: Philip E. GillWinter Quarter 2011Homework Assignment #3 Due Friday January 28, 2011 Remember that the first midterm exam will be held in class on Wednesday, January 26. Starred exercises require the use of Matla

UCSD - MATH - 171A

Math 171A: Linear ProgrammingInstructor: Philip E. GillWinter Quarter 2011Homework Assignment #4 Due Friday February 4, 2011 Starred exercises require the use of Matlab. Exercise 4.1. Suppose that the constant vector c is such that cTp 0 for all p such

UCSD - MATH - 171A

Math 171A: Linear ProgrammingInstructor: Philip E. GillWinter Quarter 2011Homework Assignment #5 Due Friday February 11, 2011 Starred exercises require the use of Matlab. Exercise 5.1. Consider the set of inequality constraints Ax b, where 1 1 4 0 3 1