Linear Programming Motivation: The Diet Problem

Transcription

1 Agenda We ve done Greedy Method Divide and Conquer Dynamic Programming Network Flows & Applications NP-completeness Now Linear Programming and the Simplex Method Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Linear Programming Motivation: The Diet Problem Setting n foods (beef, apple, potato chips, pho, bún bò, etc.) m nutritional elements (vitamins, calories, etc.) each gram of jth food contains a ij units of nutritional element i a good meal needs b i units of nutritional element i each gram of jth food costs c j Objective design the most economical meal yet dietarily sufficient (Halliburton must solve this problem!) Hung Q. Ngo (SUNY at Buffalo) CSE / 59

2 The Diet Problem as a Linear Program Let x j be the weight of food j in a dietarily sufficient meal. min c 1 x 1 + c 2 x c n x n subject to a 11 x 1 + a 12 x a 1n x n b 1 a 21 x 1 + a 22 x a 2n x n b a m1 x 1 + a m2 x a mn x n b m x j 0, j = 1,..., n, Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Linear Programming Motivation: The Max-Flow Problem Maximize the value of f: val(f) = e=(s,v) E Subject to 0 f e c e, e E f e f e = 0, e=(u,v) E e=(v,w) E f e v s, t Hung Q. Ngo (SUNY at Buffalo) CSE / 59

3 Formalizing the Linear Programming Problem Linear objective function max or min 8 3 x 1 + 2x 2 + x 3 6x 4 + x 5 Linear constraints, can take many forms Inequality constraints 3x 1 + 4x 5 2x 6 3 2x 1 + 2x 2 + x 3 0 Equality constraints x 2 x 4 + x 3 = 3 Non-negativity constraints (special case of inequality) x 1, x 5, x 7 0 Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Some notational conventions All vectors are column vectors c 1 c n x 1 x 2 c 2 c =., x =., A = x n a 11 a a 1n a 21 a a 2n , b = a m1 a m2... a mn b 1 b 2. b m. Hung Q. Ngo (SUNY at Buffalo) CSE / 59

4 Linear Program: Standard Form min / max c 1 x 1 + c 2 x c n x n subject to a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2 or, in matrix notations, =. a m1 x 1 + a m2 x a mn x n = b m x j 0, j = 1,..., n, min / max { c T x Ax = b, x 0 } Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Linear Program: Canonical Form min Version min c 1 x 1 + c 2 x c n x n subject to a 11 x 1 + a 12 x a 1n x n b 1 a 21 x 1 + a 22 x a 2n x n b 2 or, in matrix notations, a m1 x 1 + a m2 x a mn x n b m x j 0, j = 1,..., n, min { c T x Ax b, x 0 } Hung Q. Ngo (SUNY at Buffalo) CSE / 59

5 Linear Program: Canonical Form max Version max c 1 x 1 + c 2 x c n x n subject to a 11 x 1 + a 12 x a 1n x n b 1 a 21 x 1 + a 22 x a 2n x n b 2 or, in matrix notations, a m1 x 1 + a m2 x a mn x n b m x j 0, j = 1,..., n, max { c T x Ax b, x 0 } Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Conversions Between Forms of Linear Programs max c T x = min( c) T x j a ijx j = b i is equivalent to j a ijx j b i and j a ijx j b i. j a ijx j b i is equivalent to j a ijx j b i j a ijx j b i is equivalent to j a ijx j + s i = b i, s i 0. The variable s i is called a slack variable. When x j 0, replace all occurrences of x j by x j, and replace x j 0 by x j 0. When x j is not restricted in sign, replace it by (u j v j ), and u j, v j 0. Hung Q. Ngo (SUNY at Buffalo) CSE / 59

6 Example of Converting Linear Programs Write min x 1 x 2 + 4x 3 subject to 3x 1 x 2 = 3 x 2 + 2x 4 4 x 1 + x 3 3 x 1, x 2 0 in standard (min / max) form and canonical (min / max) form. Hung Q. Ngo (SUNY at Buffalo) CSE / 59 LP Geometry: Example 1 max 2x + y subject to 2x + y 2 5x + 3y 15 x + y 4 x 0, y 0 Hung Q. Ngo (SUNY at Buffalo) CSE / 59

7 Example 1 Feasible Region Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Example 1 Objective Function Hung Q. Ngo (SUNY at Buffalo) CSE / 59

8 LP Geometry: Example 2 max 2x + y subject to 2x + 3y 8 8x + 3y 12 4x + 3y 24 Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Example 2 Feasible Region Hung Q. Ngo (SUNY at Buffalo) CSE / 59

9 Example 2 Objective Function Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Half Space and Hyperplane Each inequality a T x b defines a half-space. a T x b a a T x b Each equality a T x = b defines a hyperplane. Hung Q. Ngo (SUNY at Buffalo) CSE / 59

10 Polyhedron, Vertices, Direction of Optimization A polyhedron is the set of points x satisfying Ax b (or equivalently A x b ) c T x = d 1 d 2 < d 1 c T x is improved moving along this direction vertex c c T x = d 2 Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Linear Programming Duality: A Motivating Example max 3x 1 + 2x 2 + 4x 3 subject to x 1 + x 2 + 2x 3 4 2x 1 + 3x 3 5 4x 1 + x 2 + 3x 3 7 x 1, x 2, x 3 0. Someone claims x T = [ ] is optimal with cost 9. Feasibility is easy. How could we verify optimality? We ask the person for a proof! Hung Q. Ngo (SUNY at Buffalo) CSE / 59

11 Her Proof of the Optimality of x Done! 5 3 ( x 1 + x 2 + 2x 3 ) ( 2x 1 + 3x 3 ) ( 4x 1 + x 2 + 3x 3 ) Ma ia hii Ma ia huu Ma ia hoo Ma ia haha... 3x 1 + 2x 2 + 4x 3 9 Hung Q. Ngo (SUNY at Buffalo) CSE / 59 OK, How Did... I do that? Beside listening to Numa Numa, find non-negative multipliers y 1 ( x 1 + x 2 + 2x 3 ) 4y 1 y 2 ( 2x 1 + 3x 3 ) 5y 2 y 3 ( 4x 1 + x 2 + 3x 3 ) 7y 3 (y 1 +2y 2 +4y 3 )x 1 +(y 1 +y 3 )x 2 +(2y 1 +3y 2 +3y 3 )x 3 (4y 1 +5y 2 +7y 3 ) Want LHS to be like 3x 1 + 2x 2 + 4x 3. Thus, as long as y 1 + 2y 2 + 4y 3 3 y 1 + y 3 2 2y 1 + 3y 2 + 3y 3 4 y 1, y 2, y 3 0. we have 3x 1 + 2x 2 + 4x 3 4y 1 + 5y 2 + 7y 3 Hung Q. Ngo (SUNY at Buffalo) CSE / 59

12 How to Get the Best Multipliers Answer: minimize the upper bound. min 4y 1 + 5y 2 + 7y 3 y 1 + 2y 2 + 4y 3 3 y 1 + y 3 2 2y 1 + 3y 2 + 3y 3 4 y 1, y 2, y 3 0. Hung Q. Ngo (SUNY at Buffalo) CSE / 59 What We Have Just Shown If x is feasible for the Primal Program max 3x 1 + 2x 2 + 4x 3 subject to x 1 + x 2 + 2x 3 4 2x 1 + 3x 3 5 4x 1 + x 2 + 3x 3 7 x 1, x 2, x 3 0. and y is feasible for the Dual Program min 4y 1 + 5y 2 + 7y 3 y 1 + 2y 2 + 4y 3 3 y 1 + y 3 2 2y 1 + 3y 2 + 3y 3 4 y 1, y 2, y 3 0. then 3x 1 + 2x 2 + 4x 3 4y 1 + 5y 2 + 7y 3 Hung Q. Ngo (SUNY at Buffalo) CSE / 59

13 Primal-Dual Pairs - Canonical Form min c T x (primal/dual program) subject to Ax b x 0 max b T y (dual/primal program) subject to A T y c y 0. Note The dual of the dual is the primal! Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Primal-Dual Pairs - Standard Form min / max c T x (primal program) subject to Ax = b x 0 max / min b T y (dual program) subject to A T y c no non-negativity restriction!. Hung Q. Ngo (SUNY at Buffalo) CSE / 59

14 General Rules for Writing Dual Programs Maximization problem Minimization problem Constraints Variables ith constraint ith variable 0 ith constraint ith variable 0 ith constraint = ith variable unrestricted Variables Constraints jth variable 0 jth constraint jth variable 0 jth constraint jth variable unrestricted jth constraint = Table: Rules for converting between primals and duals. Note The dual of the dual is the primal! Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Weak Duality Canonical Form Version Consider the following primal-dual pair in canonical form Theorem (Weak Duality) Primal LP: min{c T x Ax b, x 0}, Dual LP: max{b T y A T y c, y 0}. Suppose x is primal feasible, and y is dual feasible for the LPs defined above, then c T x b T y. Corollary If x is an primal-optimal and y is an dual-optimal, then c T x b T y. Corollary If x is primal-feasible, y is dual-feasible, and c T x = b T y, then x and y are optimal for their respective programs. Hung Q. Ngo (SUNY at Buffalo) CSE / 59

15 Weak Duality Standard Form Version Consider the following primal-dual pair in standard form Primal LP: min{c T x Ax = b, x 0}, Dual LP: Theorem (Weak Duality) max{b T y A T y c}. Suppose x is primal feasible, and y is dual feasible for the LPs defined above, then c T x b T y. Corollary If x is an primal-optimal and y is an dual-optimal, then c T x b T y. Corollary If x is primal-feasible, y is dual-feasible, and c T x = b T y, then x and y are optimal for their respective programs. Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Strong Duality Theorem (Strong Duality) If the primal LP has an optimal solution x, then the dual LP has an optimal solution y such that c T x = b T y. Dual Feasible Infeasible Optimal Unbounded Feasible Optimal X Nah Nah Primal Unbounded Nah Nah X Infeasible Nah X X Hung Q. Ngo (SUNY at Buffalo) CSE / 59

16 The Diet Problem Revisited The dual program for the diet problem: max b 1 y 1 + b 2 y b m y m subject to a 11 y 1 + a 21 y a m1 y m c 1 a 12 y 1 + a 22 y a 2n y m c a 1n y 1 + a 2n y a mn y m c n y j 0, j = 1,..., m, (Possible) Interpretation: y i is the price per unit of nutrient i that a whole-seller sets to manufacture different types of foods. Hung Q. Ngo (SUNY at Buffalo) CSE / 59 The Max-Flow Problem Revisited The dual program for the Max-Flow LP Formulation: min c uv y uv uv E subject to y uv z u + z v 0 uv E z s = 1 z t = 0 y uv 0 uv E Theorem (Max-Flow Min-Cut) Maximum flow value equal minimum cut capacity. Proof. Let (y, z ) be optimal to the dual above. Set W = {v z v 1}, then total flow out of W is equal to cap(()w, W ). Hung Q. Ngo (SUNY at Buffalo) CSE / 59

17 The Simplex Method: High-Level Overview Consider a linear program min{c T x x P }, P is a polyhedron 1 Find a vertex of P, if P is not empty (the LP is feasible) 2 Find a neighboring vertex with better cost If found, then repeat step 2 Otherwise, either report unbounded or optimal solution Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Questions 1 When is P not empty? 2 When does P have a vertex? (i.e. P is pointed) 3 What is a vertex, anyhow? 4 How to find an initial vertex? 5 What if no vertex is optimal? 6 How to find a better neighboring vertex 7 Will the algorithm terminate? 8 How long does it take? Hung Q. Ngo (SUNY at Buffalo) CSE / 59

18 3. What is a vertex, anyhow? u w y +y vertex v Many ways to define a vertex v: v P a vertex iff y 0 with v + y, v y P v P a vertex iff u w such that v = (u + w)/2 v P a vertex iff it s the unique intersection of n independent faces Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Questions 1 When is P not empty? 2 When does P have a vertex? (i.e. P is pointed) 3 What is a vertex, anyhow? 4 How to find an initial vertex? 5 What if no vertex is optimal? 6 How to find a better neighboring vertex 7 Will the algorithm terminate? 8 How long does it take? Hung Q. Ngo (SUNY at Buffalo) CSE / 59

19 2. When is P pointed? Question Define a polyhedron which has no vertex? Lemma P is pointed iff it contains no line Lemma P = {x Ax = b, x 0}, if not empty, always has a vertex. Lemma v P = {x Ax = b, x 0} is a vertex iff the columns of A corresponding to non-zero coordinates of v are linearly independent Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Questions 1 When is P not empty? 2 When does P have a vertex? (i.e. P is pointed) 3 What is a vertex, anyhow? 4 How to find an initial vertex? 5 What if no vertex is optimal? 6 How to find a better neighboring vertex 7 Will the algorithm terminate? 8 How long does it take? Hung Q. Ngo (SUNY at Buffalo) CSE / 59

20 5. What if no vertex is optimal? Lemma Let P = {x Ax = b, x 0}. If min { c T x x P } is bounded (i.e. it has an optimal solution), then for all x P, there is a vertex v P such that c T v c T x. Theorem The linear program min{c T x Ax = b, x 0} either 1 is infeasible, 2 is unbounded, or 3 has an optimal solution at a vertex. Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Questions 1 When is P not empty? 2 When does P have a vertex? (i.e. P is pointed) 3 What is a vertex, anyhow? 4 How to find an initial vertex? 5 What if no vertex is optimal? 6 How to find a better neighboring vertex 7 Will the algorithm terminate? 8 How long does it take? Hung Q. Ngo (SUNY at Buffalo) CSE / 59

21 6. How to find a better neighboring vertex The answer is the core of the Simplex method This is basically one iteration of the method Consider a concrete example: max 3x 1 + 2x 2 + 4x 3 subject to x 1 + x 2 + 2x 3 4 2x 1 + 3x 3 5 4x 1 + x 2 + 3x 3 7 x 1, x 2, x 3 0. Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Sample execution of the Simplex algorithm Converting to standard form max 3x 1 +2x 2 +4x 3 subject to x 1 +x 2 +2x 3 +x 4 = 4 2x 1 +3x 3 +x 5 = 5 4x 1 +x 2 +3x 3 +x 6 = 7 x 1, x 2, x 3, x 4, x 5, x 6 0. x = [ ] T is a vertex! Define B = {4, 5, 6}, N = {1, 2, 3}. The variables x i, i N are called free variables. The x i with i B are basic variables. How does one improve x? Increase x 3 as much as possible! (x 1 or x 2 works too.) Hung Q. Ngo (SUNY at Buffalo) CSE / 59

22 Sample execution of the Simplex algorithm x 3 can only be at most 5/3, forcing x 4 = 2/3, x 5 = 0, x 5 = 2 x T = [ 0 0 5/3 2/3 0 2 ] is the new vertex (why?!!!) The new objective value is 20/3 x 3 enters the basis B, x 5 leaves the basis B = {3, 4, 6}, N = {1, 2, 5} Rewrite the linear program max 3x 1 +2x 2 +4x 3 subject to 1 3 x 1 +x 2 +x 4 = x 1 +x x 5 = 5 3 2x 1 +x 2 +x 6 = 2 x 1, x 2, x 3, x 4, x 5, x 6 0. Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Sample execution of the Simplex algorithm We also want the objective function to depend only on the free variables: ( 5 3x 1 + 2x 2 + 4x 3 = 3x 1 + 2x x 1 1 ) 3 x 5 The linear program is thus equivalent to = 1 3 x 1 + 2x x max 3 x 1 +2x x subject to 1 3 x 1 +x 2 +x 4 = x 1 +x x 5 = 5 3 2x 1 +x 2 +x 6 = 2 Increase x 2 to 2/3, so that x 2 enters, x 4 leaves. x 1, x 2, x 3, x 4, x 5, x 6 0. Hung Q. Ngo (SUNY at Buffalo) CSE / 59

23 Sample execution of the Simplex algorithm max x 1 2x subject to 1 3 x 1 +x 2 +x 4 = x 1 +x x 5 = x 1 x x 5 +x 6 = 4 3 x 1, x 2, x 3, x 4, x 5, x 6 0. At this point, only x 1 to increase. If all its coefficients are non-positive (like 1/3 above), then the LP is unbounded Fortunately, this is not the case here Increase x 1 to 4/7, so that x 1 enters, x 6 leaves. Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Sample execution of the Simplex algorithm max 11 7 x x x subject to +x x x x 6 = 6 7 Now, x 5 enters again, x 3 leaves. +x x x x 6 = 9 7 x x x x 6 = 4 7 x 1, x 2, x 3, x 4, x 5, x 6 0. Hung Q. Ngo (SUNY at Buffalo) CSE / 59

24 Sample execution of the Simplex algorithm max 1 3 x x x subject to +x x x x 6 = x x 4 +x x 6 = 3 x x x x 6 = 1 x 1, x 2, x 3, x 4, x 5, x 6 0. Yeah! No more improvement is possible. We have reached the optimal vertex v = [ ] T. The optimal cost is 9. Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Questions 1 When is P not empty? 2 When does P have a vertex? (i.e. P is pointed) 3 What is a vertex, anyhow? 4 How to find an initial vertex? 5 What if no vertex is optimal? 6 How to find a better neighboring vertex 7 Will the algorithm terminate? 8 How long does it take? Hung Q. Ngo (SUNY at Buffalo) CSE / 59

25 7&8 Termination and Running Time Termination There are finitely many vertices ( ( n m) ) Terminating = non-cycling, i.e. never come back to a vertex Many cycling prevention methods: perturbation method, lexicographic rule, Bland s pivoting rule, etc. Bland s pivoting rule: pick smallest possible j to leave the basis, then smallest possible i to enter the basis Running time Klee & Minty (1969) showed that Simplex could take exponential time Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Summary: Simplex with Bland s Rule 1 Start from a vertex v of P. 2 Determine B and N; Let y T B = ct B A 1 B. 3 If ( c T N yt B a j) 0, then vertex v is optimal. Moreover, 4 Else, let c T v = c T Bv B + c T Nv N = c T B ( A 1 B a j k ( A 1 B b A 1 B A Nv N ) + c T N v N j = min { j N : ( c j y T Ba j ) < 0 }. 5 If A 1 B a j 0, then report unbounded LP and Stop! 6 Otherwise, pick smallest k B such that ( A 1 B a ) j > 0 and that k { } (A 1 B b) k (A 1 ) B = min b) i ( ) : i B, A 1 B a ) j i > 0. ( A 1 B a j 7 x k leaves, x j enters: B = B {j} {k}, N = N {k} {j}. Go back to step 3. Hung Q. Ngo (SUNY at Buffalo) CSE / 59 i

26 By Product: Strong Duality Theorem (Strong Duality) If the primal LP has an optimal solution x, then the dual LP has an optimal solution y such that c T x = b T y. Proof. Suppose Simplex returns vertex x (at B and N) Recall yb T = ct B A 1 B, then ct x = yb T b [ ] [ ] A T A T y B = B cb y B = A T N y B A T N [ cb c N ] = c. Set y = y T B. Done! Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Questions 1 When is P not empty? 2 When does P have a vertex? (i.e. P is pointed) 3 What is a vertex, anyhow? 4 How to find an initial vertex? 5 What if no vertex is optimal? 6 How to find a better neighboring vertex 7 Will the algorithm terminate? 8 How long does it take? Hung Q. Ngo (SUNY at Buffalo) CSE / 59

27 1&4 Feasibility and the Initial Vertex In P = {x Ax = b, x 0}, we can assume b 0 (why?). Let A = [ A I ] Let P = {z A z = b, z 0}. A vertex of P is z = [0,..., 0, b 1,..., b m ] T P is feasible iff the following LP has optimum value 0 { m } min z n+i z P i=1 From an optimal vertex z, ignore the last m coordinates to obtain a vertex of P Hung Q. Ngo (SUNY at Buffalo) CSE / 59 Questions 1 When is P not empty? 2 When does P have a vertex? (i.e. P is pointed) 3 What is a vertex, anyhow? 4 How to find an initial vertex? 5 What if no vertex is optimal? 6 How to find a better neighboring vertex 7 Will the algorithm terminate? 8 How long does it take? Hung Q. Ngo (SUNY at Buffalo) CSE / 59