Lecture 9 Tuesday, 4/20/10. Linear Programming
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1 UMass Lowell Computer Science Analysis of Algorithms Prof. Karen Daniels Spring, 2010 Lecture 9 Tuesday, 4/20/10 Linear Programming 1
2 Overview Motivation & Basics Standard & Slack Forms Formulating Problems as Linear Programs Simplex Algorithm Example High Level Algorithm Correctness Roadmap Key Concepts Initial Basic Feasible Solution Duality More examples 2
3 Motivation & Basics 3
4 Motivation: A Political Problem Goal: Win election by winning majority of votes in each region. 100,000 voters 200,000 voters 50,000 voters Thousands of voters who could be won with $1,000 of ads related to a policy Subgoal: Win majority of votes in each region while minimizing advertising cost. 4
5 Motivation: A Political Problem (continued) This search space is 4-dimensional. urban suburban rural Thousands of voters representing majority. 5
6 General Linear Programs Linear function real numbers real-valued variables = Linear constraints Linear inequalities 6
7 Overview of Linear Programming Different from voter example: this example has 2-dimensional search space Objective function Convex feasible region: intersection ti of half-planes Objective value 7
8 Overview of Linear Programming More examples (on board) of 2-dimensional search space: -Infeasible example in which constraints conflict -Unbounded example
9 Standard & Slack Forms 9
10 Standard Form objective function constraints 10
11 Standard Form (compact) n-dimensional vectors mxn matrix m-dimensional vector Can specify linear program in standard form by (A,b,c). 11
12 Converting to Standard Form 12
13 Converting to Standard Form (continued) Transforming minimization imization to maximization Negate coefficients 13
14 Converting to Standard Form (continued) Giving each variable a non-negativity negativity constraint -3(x 2 x 2 ) x 2 has no non-negativity constraint New non-negativity constraints If x j has no non-negativity negativity constraint, replace each occurrence of x j with x j x j. 14
15 Converting to Standard Form (continued) Transforming equality constraints to inequality constraints 15
16 Converting to Standard Form (continued) Changing sense of an inequality constraint (Rename variables for notational consistency.) Rationale: 16
17 Converting Linear Programs into Slack Form for algorithmic ease, transform all constraints except non-negativity negativity ones into equalities slack variable for inequality constraint: define slack instead of s non-basic variables basic variables 17
18 Converting Linear Programs into Slack Form (continued) objective function happens to be empty in this case 18
19 Converting Linear Programs into Slack Form (continued) Compact Form: (N, B, A, b, c, v) set of indices of basic variables set of indices of non-basic variables Slack Form Example Compact Form negative of slack form coefficients basic variables 19
20 Formulating Problems as Linear Programs 20
21 Shortest Paths Single-pair i shortest t path: minimize i i distance from source s to sink t. Use Bellman-Ford idea: translate relaxation into a constraint. Recall: δ ( s, v) δ ( s, u) + w( u, v) where δ is shortest path weight Can we replace maximize with minimize here? Why or why not?, 2 nd edition 21
22 Capacity constraints t Flow conservation Maximum Flow maximize subject to v V f uv v V f sv f v V f c ( u, v ) for each u, v V f uv f vu = v V f uv for vs each 0 for each u, v V u V { s, t}, 3 rd edition 22
23 Minimum Cost Flow minimize subject to Capacity constraints Flow conservation Flow target d is prespecified. f uv v V v V f uv f ( u, v) E a( u, v) f uv c ( u, v ) for each u, v V vu v V f uv f sv fvs = d v V 0 for each u, v V = 0 for each u V { s, t} cost = 4 cost = 7 cost = 7 Flow target d is cost = 3 prespecified here as d=4. cost = 3 cost = 3 cost = 10 Total weighted flow Total cost weighted = 27 flow cost = 27, 3 rd edition 23
24 Multicommodity Flow subject to Flow conservation v V minimize Capacity constraints f ivu v V Flow target d i is prespecified for each i. k 0 k is number of commodities d i is target flow value for commodity i s i is source for commodity i t i is sink for commodity i f iuv c( u, v) for each u, v V f i=1 iuv v V = 0 for each i = 1, 2,...k and each u V { si, ti} f is v fivs = di for each i = i i v V f iuv 0 for each u, v V and for each i = 1,2,... k 1,2,... k The only known polynomial-time algorithm for this problem expresses it as a linear program and then solves it with a polynomial-time linear-programming algorithm., 3 rd edition 24
25 Simplex Method 25
26 Simplex algorithm Solving a Linear Program Geometric interpretation Visit some vertices on the boundary of the simplex representing the convex feasible region Transforms set of inequalities using process similar to Gaussian elimination Run time not polynomial in worst case 1972 exponential time result often very fast in practice sparse matrix techniques provide speed up Ellipsoid method Iterative approach creating sequence of ellipsoids: holds current point as center of ellipsoid that holds a feasible point, if one exists. Cut off part of ellipsoid violated by some constraint. Run time polynomial slow in practice Interior Point methods Run time polynomial for large inputs, performance can be competitive with simplex method Moves through interior of feasible region 26
27 Simplex Algorithm: Example Basic Solution Standard Form Slack Form Basic Solution: ( 6 x1, x2 Kx ) = (0,0,0,30,24,36) Basic Solution: set each nonbasic variable to 0. 27
28 Simplex Algorithm: Example Reformulating the LP Model Main Idea: In each iteration, reformulate the LP model so basic solution has larger objective value (since we re maximizing). Select a nonbasic variable whose objective coefficient is positive (since we re maximizing): x 1 Increase its value as much as possible. Identify tightest constraint on increase. For basic variable x 6 of that constraint, swap role with x 1. Rewrite other equations with x 6 on RHS. new objective value entering variable PIVOT leaving variable 28
29 Simplex Algorithm: Example Reformulating the LP Model Next Iteration: select x 3 as entering variable. entering variable new objective value ( 6 x1, x2 Kx ) = (33/ 4,0,3/ 2,69 / 4,0,0) PIVOT leaving variable New Basic Solution: ( x, x Kx6) (33/ 4,0,3/ 2,69 / 4,0,0 0 ) ( 1 2 x6 = 29
30 Simplex Algorithm: Example Reformulating the LP Model Next Iteration: select x 2 as entering variable. entering variable new objective value Stop since all coefficients are negative. ( 6 x1, x2 Kx ) = (8,4,0,18,0,0) PIVOT leaving variable New Basic Solution: ( x, x K x ) = (8,4,0,18,0,0 0 0 ) ( = 30
31 Simplex Algorithm: Pivoting leaving variable entering variable Rewrite the equation that has x l on LHS to have x e on LHS Update remaining equations by substituting RHS of new equation for each occurrence of x e. Do the same for objective function. Update sets of nonbasic, basic variables. 31
32 Simplex Algorithm: Pseudocode initial basic solution to be defined later (detects infeasibility) detects unboundedness optimal solution 32
33 Correctness: Roadmap (Key Pieces) Lemma 29.1: Pivot results Lemma 29.11: L aux Lemma 29.12: Infeasibility detection Lemma 29.2: Lemma 29.3: Algebraic lemma Lemma 29.8: Lemma 29.4: Slack form uniqueness Lemma 29.6: Tie-breaking Basic solution feasible -> if SIMPLEX finds solution it is feasible; if reports unbounded, then model is unbounded Lemma 29.5: Iteration bound, cycling n + m # variables m # basic variables Lemma 29.7: Basic solution feasible -> SIMPLEX either reports unbounded or finds iterations ti m Weak LP duality Corollary 29.9: Conditions for which feasible solutions for primal, (max) and dual (min) programs are optimal feasible; if reports feasible solution in n + m Theorem 29.10: Theorem 29.13: Fundamental Theorem of Linear Programming g LP duality: SIMPLEX primal result is optimal & dual is optimal * analogous to max flow/ min cut idea For LP model in standard form, either: 1. exists optimal solution with finite objective function value & SIMPLEX returns one, or 2. infeasible & SIMPLEX returns INFEASIBLE, or 3. unbounded & SIMPLEX returns UNBOUNDED (Lemmas 29.2 & 29.7) 33
34 Proof of Correctness Key Concepts Initial Basic Feasible Solution 34
35 Finding an Initial Solution An LP model whose initial basic solution is not feasible Board work for picture, slack form and initial basic solution. 35
36 Finding an Initial Solution (continued) Auxiliary LP model L aux : max -x 0 L aux in standard but not slack form x 0 0 x0 0 L feasible -> 0 optimal for L aux L feasible <- 0 optimal for L aux 0 36
37 Finding an Initial Solution (continued) most negative see p. 888 for details (also example in next slides) 3 rd edition 37
38 Finding an Initial Solution (continued) Original LP model L aux L aux in slack form L aux Basic (infeasible) Solution: ( 4 x0, x1, x2, x3, x ) = (0,0,0,2, 4) 38
39 Finding an Initial Solution (continued) Entering entering Leaving (most restrictive) PIVOT Entering most leaving Leaving (most negative negative, restrictive) (restrictive) PIVOT L aux Basic (feasible) Solution: ( x0, x1, x2, x3, x4 ) = Why is this feasible for L aux? (4,0,0,6,0) 0 0) Why can we stop here? Feasible Basic Solution: (0, 4/5, 14/5, 0) 39
40 Finding an Initial Solution (continued) See textbook for proof (p. 890). 40
41 Proof of Correctness Key Concepts Duality 41
42 Linear Programming Duality x variables go away max becomes min sense changes RHS coefficients swap places with objective function coefficients y variables appear 42
43 Duality Example 43
44 Weak Linear Programming Duality Any feasible solution to primal LP has value no greater than that of any feasible solution to the dual LP. standard form
45 Weak Linear Programming Duality (continued) maximized minimized 45
46 Finding a Dual Solution Finding a dual solution whose value is equal to that of an optimal primal solution Suppose the last slack form of the primal is: z = v'+ j N c' j x j x i = b' i a' ' ij x j for i B. j N Then, to produce an optimal dual solution, we set: y j = c' 0 n ( n + i) + i if N, otherwise. ( 29.91) 3 rd edition 46
47 Optimality Theorem (Linear-Programming P i Duality). ) Suppose that SIMPLEX returns values x = ( x1, x2, K, x n ) for the primal linear program (A, b, c). Let N and B denote the non basic and basic variables for the final slack form, let c denote the coefficient in the final slack form, and let y = ( y1, y2, K, y m ) be defined by (29.91). Then x is an optimal solution the primal program, y is an optimal solution to the dual n m program and c j x j = b i y i, j =11 i=1 1 Proof: see text (p ). 47
48 Correctness: Roadmap (Key Pieces) Lemma 29.1: Pivot results Lemma 29.11: L aux Lemma 29.12: Infeasibility detection Lemma 29.2: Lemma 29.3: Algebraic lemma Lemma 29.8: Lemma 29.4: Slack form uniqueness Lemma 29.6: Tie-breaking Basic solution feasible -> if SIMPLEX finds solution it is feasible; if reports unbounded, then model is unbounded Lemma 29.5: Iteration bound, cycling n + m # variables m # basic variables Lemma 29.7: Basic solution feasible -> SIMPLEX either reports unbounded or finds iterations ti m Weak LP duality Corollary 29.9: Conditions for which feasible solutions for primal, (max) and dual (min) programs are optimal feasible; if reports feasible solution in n + m Theorem 29.10: Theorem 29.13: Fundamental Theorem of Linear Programming g LP duality: SIMPLEX primal result is optimal & dual is optimal * analogous to max flow/ min cut idea For LP model in standard form, either: 1. exists optimal solution with finite objective function value & SIMPLEX returns one, or 2. infeasible & SIMPLEX returns INFEASIBLE, or 3. unbounded & SIMPLEX returns UNBOUNDED (Lemmas 29.2 & 29.7) 48
49 Beyond Chapter 29 49
50 Some Variations on Linear Programming Integer Programming: Linear constraints and objective Variables are restricted to integer values. Feasibility is NP complete. Mixed Integer Programming: Linear constraints and objective Some variable(s) are restricted to integer values. Some variables are not restricted to integers. Quadratic Programming: Linear constraints Quadratic objective function Real valued variables Convex Programming Generalization of Linear Programming Minimize a convex function, subject to convex constraints Interior point methods can be applied if the functions satisfy additional properties. Reference: A Tutorial on Convex Optimization by Hindi Increasingly being used. 50
51 A Hierarchy of Related Types of Mathematical Programming 51
52 See docs portion of web site for additional information on Karp s work. (courtesy of Nate) 52
53 My Applied Mathematical Programming Research Lagrangian Relaxation Apparel trim placement (Grinde & Daniels 2000) Polygon covering (Daniels et al. 2003) Box covering (England, Daniels 2008) Mixed Integer Programming Dynamic Channel assignment (Liu, Daniels, Chandra 2001, 2004) Quadratic Programming Support vector clustering enhancements (Lee, Daniels 2005, 2006) See 53
54 Linear Programming Resource Elementary Linear Programming with Applications (2 nd edition), by Kolman and Beck, Academic Press,
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