Introduction to integer programming II

Introduction to integer programming II Martin Branda Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Computational Aspects of Optimization Martin Branda (KPMS MFF UK) 2017-03-20 1 / 39

Obsah Introduction to computational complexity 1 Introduction to computational complexity 2 3 Duality 4 Dynamic programming Martin Branda (KPMS MFF UK) 2017-03-20 2 / 39

Introduction to computational complexity Introduction to complexity theory Wolsey (1998): Consider decision problems having YES NO answers. Optimization problem max x M ct x can be replaced by (for some k integral) Is there an x M with value c T x k? For a problem instance X, the length of the input L(X ) is the length of the binary representation of a standard representation of the instance. Instance X = {c, M}, X = {c, M, k}. Martin Branda (KPMS MFF UK) 2017-03-20 3 / 39

Introduction to computational complexity Example: Knapsack decision problem For an instance { n } n X = c i x i k, a i x i b, x {0, 1} n, the length of the input is i=1 i=1 L(X ) = n n log c i + log a i + log b + log k i=1 i=1 Martin Branda (KPMS MFF UK) 2017-03-20 4 / 39

Introduction to computational complexity Running time Definition f A (X ) is the number of elementary calculations required to run the algorithm A on the instance X P. Running time of the algorithm A f A (l) = sup{f A (X ) : L(X ) = l}. X An algorithm A is polynomial for a problem P if f A (l) = O(l p ) for some p N. Martin Branda (KPMS MFF UK) 2017-03-20 5 / 39

Introduction to computational complexity Classes N P and P Definition N P (Nondeterministic Polynomial) is the class of decision problems with the property that: for any instance for which the answer is YES, there is a polynomial proof of the YES. P is the class of decision problems in N P for which there exists a polynomial algorithm. N P may be equivalently defined as the set of decision problems that can be solved in polynomial time on a non-deterministic Turing machine 1. 1 NTM writes symbols one at a time on an endless tape by strictly following a set of rules. It determines what action it should perform next according to its internal state and what symbol it currently sees. It may have a set of rules that prescribes more than one action for a given situation. The machine branches into many copies, each of which follows one of the possible transitions leading to a computation tree. Martin Branda (KPMS MFF UK) 2017-03-20 6 / 39

Introduction to computational complexity Alan Turing The Imitation Game (2014) Martin Branda (KPMS MFF UK) 2017-03-20 7 / 39

Introduction to computational complexity Polynomial reduction and the class N PC Definition If problems P, Q N P, and if an instance of P can be converted in polynomial time to an instance of Q, then P is polynomially reducible to Q. N PC, the class of N P-complete problems, is the subset of problems P N P such that for all Q N P, Q is polynomially reducible to P. Proposition: Suppose that problems P, Q N P. If Q P and P is polynomially reducible to Q, then P P. If P N PC and P is polynomially reducible to Q, then Q N PC. Proposition: If P N PC, then P = N PC. Martin Branda (KPMS MFF UK) 2017-03-20 8 / 39

Introduction to computational complexity Open question & Euler diagram Is P = N P? Martin Branda (KPMS MFF UK) 2017-03-20 9 / 39

Introduction to computational complexity N P-hard optimization problems Definition An optimization problem for which the decision problem lies in N PC is called N P-hard. Martin Branda (KPMS MFF UK) 2017-03-20 10 / 39

Introduction to computational complexity Simplex algorithm Klee Minty (1972) example: n max 10 n j x j j=1 i 1 s.t. 2 10 i j x j + x i 100 i 1, i = 1,..., n, j=1 x j 0, j = 1,..., n. (1) Can be easily reformulated in the standard form. The Simplex algorithm takes 2 n 1 pivot steps, i.e. it is not polynomial in the worst case. Martin Branda (KPMS MFF UK) 2017-03-20 11 / 39

Obsah 1 Introduction to computational complexity 2 3 Duality 4 Dynamic programming Martin Branda (KPMS MFF UK) 2017-03-20 12 / 39

Basic idea: DIVIDE AND RULE Let M = M 1 M 2 M r be a partitioning of thefeasibility set and let f j = min x M j f (x). Then min f (x) = min f j. x M j=1,...,r Martin Branda (KPMS MFF UK) 2017-03-20 13 / 39

General principles: Solve LP problem without integrality only. Branch using additional constraints on integrality: x i x i, x i x i + 1. Cut inperspective branches before solving (using bounds on the optimal value). Martin Branda (KPMS MFF UK) 2017-03-20 14 / 39

General principles: Solve only LP problems with relaxed integrality. Branching: if an optimal solution is not integral, e.g. ˆx i, create and save two new problems with constraints x i ˆx i, x i ˆx i. Bounding ( different cutting): save the objective value of the best integral solution and cut all problems in the queue created from the problems with higher optimal values 2. Exact algorithm.. 2 Branching cannot improve it. Martin Branda (KPMS MFF UK) 2017-03-20 15 / 39

P. Pedegral (2004). Introduction to optimization, Springer-Verlag, New York. Martin Branda (KPMS MFF UK) 2017-03-20 16 / 39

0. f min =, x min =, list of problems P = Solve LP-relaxed problem and obtain f, x. If the solution is integral, STOP. If the problem is infeasible or unbounded, STOP. 1. BRANCHING: There is xi for some k N: basic non-integral variable such that k < x i < k + 1 Add constraint x i k to previous problem and put it into list P. Add constraint x i k + 1 to previous problem and put it into list P. 2. Take problem from P and solve it: f, x. 3. If f < f min and x is non-integral, GO TO 1. BOUNDING: If f < f min a x is integral, set f min = f a x min = x, GO TO 4. BOUNDING: If f f min, GO TO 4. Problem is infeasible, GO TO 4. 4. If P, GO TO 2. If P = a f min =, integral solution does not exist. If P = a f min <, optimal value and solution are f min, x min. Martin Branda (KPMS MFF UK) 2017-03-20 17 / 39

Better... 2./3. Take problem from list P and solve it: f, x. If for the optimal value of the current problem holds f f min, then the branching is not necessary, since by solving the problems with added branching constraints we can only increase the optimal value and obtain the same f min. Martin Branda (KPMS MFF UK) 2017-03-20 18 / 39

Algorithmic issues: Problem selection from the list P: FIFO, LIFO (depth-first search), problem with the smallest f. Selection of the branching variable xi : the highest/smallest violation of integrality OR the highest/smallest coefficient in the objective function. Martin Branda (KPMS MFF UK) 2017-03-20 19 / 39

B&B Example I min 4x 1 + 5x 2 x 1 + 4x 2 5, 3x 1 + 2x 2 7, x 1, x 2 Z +. After two iterations of the dual SIMPLEX algorithm... 4 5 0 0 x 1 x 2 x 3 x 4 5 x 2 8/10 0 1-3/10 1/10 4 x 1 18/10 1 0 2/10-4/10 112/10 0 0-7/10-11/10 Martin Branda (KPMS MFF UK) 2017-03-20 20 / 39

B&B Example I Branching means adding a cut of the form x 1 1, i.e. (α = (1, 0, 0, 0, 1), α B = (1, 0)) x 1 + x 5 = 1, x 5 0. 4 5 0 0 0 x 1 x 2 x 3 x 4 x 5 5 x 2 8/10 0 1-3/10 1/10 0 4 x 1 18/10 1 0 2/10-4/10 0 0 x 5-8/10 0 0-2/10 4/10 1 112/10 0 0-7/10-11/10 0 Dual feasible, primal infeasible run the dual simplex... Martin Branda (KPMS MFF UK) 2017-03-20 21 / 39

x 1 1 node 1 x 1 = 1 x 2 = 2 ˆf = 14 node 0 x 1 = 1.8 x 2 = 0.8 ˆf = 11.2 x 1 2 x 2 0 node 3 x 1 = 5 x 2 = 0 ˆf = 20 node 2 x 1 = 2 x 2 = 0.75 ˆf = 11.75 x 2 1 node 4 x 1 = 2 x 2 = 1 ˆf = 13 Martin Branda (KPMS MFF UK) 2017-03-20 22 / 39

B&B Example II max 23x 1 + 19x 2 + 28x 3 + 14x 4 + 44x 5 s.t. 8x 1 + 7x 2 + 11x 3 + 6x 4 + 19x 5 25, x 1, x 2, x 3, x 4, x 5 {0, 1}. Martin Branda (KPMS MFF UK) 2017-03-20 23 / 39

B&B Example II x 3 = 0 node 1 ˆf = 65.26 branching continues... node 0 ˆf = 67.45 x 3 = 1 x 2 = 0 node 3 ˆf = 65 integral node 2 ˆf = 67.28 x 2 = 1 x 1 = 0 node 5 ˆf = 63.32 pruned node 4 ˆf = 67.127 x 1 = 1 node 6 ˆf = Martin Branda (KPMS MFF UK) 2017-03-20 24 / 39

remarks If you are able to get a feasible solution quickly, deliver it to the software (solver). Branch-and-Cut: add cuts at the beginning/during B&B. Algorithm termination: (Relative) difference between a lower and an upper bound construct the upper bound (for minimization) using a feasible solution, lower bound... Martin Branda (KPMS MFF UK) 2017-03-20 25 / 39

Obsah Duality 1 Introduction to computational complexity 2 3 Duality 4 Dynamic programming Martin Branda (KPMS MFF UK) 2017-03-20 26 / 39

Duality Duality Set S(b) = {x Z n + : Ax = b} and define the value function A dual function F : R m R A general form of dual problem z(b) = min x S(b) ct x. (2) F (b) z(b), b R m. (3) max F {F (b) : s.t. F (b) z(b), b Rm, F : R m R}. (4) We call F a weak dual function if it is feasible, and strong dual if moreover F (b) = z(b). Martin Branda (KPMS MFF UK) 2017-03-20 27 / 39

Duality Duality A function F is subadditive over a domain Θ if for all θ 1 + θ 2, θ 1, θ 2 Θ. F (θ 1 + θ 2 ) F (θ 1 ) + F (θ 2 ) The value function z is subadditive over {b : S(b) }, since the sum of optimal x s is feasible for the problem with b 1 + b 2 r.h.s., i.e. ˆx 1 + ˆx 2 S(b 1 + b 2 ). Martin Branda (KPMS MFF UK) 2017-03-20 28 / 39

Duality Duality If F is subadditive, then condition F (Ax) c T x for x Z n + is equivalent to F (a j ) c j, j = 1,..., m. This is true since F (Ae j ) c T e j is the same as F (a j ) c j. On the other hand, if F is subadditive and F (a j ) c j, j = 1,..., m imply F (Ax) m F (a j )x j j=1 m c j x j = c T x. j=1 Martin Branda (KPMS MFF UK) 2017-03-20 29 / 39

Duality Duality If we set Γ m = {F : R m R, F (0) = 0, F subadditive}, then we can write a subadditive dual independent of x: max F {F (b) : s.t. F (a j) c j, F Γ m }. (5) Weak and strong duality holds. An easy feasible solution based on LP duality (= weak dual) F LP (b) = max b T y s.t. A T y c. (6) y Martin Branda (KPMS MFF UK) 2017-03-20 30 / 39

Duality Duality Complementary slackness condition: if ˆx is an optimal solution for IP, and ˆF is an optimal subadditive dual solution, then ( ˆF (a j ) c j )ˆx j = 0, j = 1,..., m. Martin Branda (KPMS MFF UK) 2017-03-20 31 / 39

Obsah Dynamic programming 1 Introduction to computational complexity 2 3 Duality 4 Dynamic programming Martin Branda (KPMS MFF UK) 2017-03-20 32 / 39

Dynamic programming Knapsack problem max s.t. n c i x i i=1 n a i x i b, i=1 x i {0, 1}. Martin Branda (KPMS MFF UK) 2017-03-20 33 / 39

Dynamic programming Dynamic programming Let a i, b be positive integers. r f r (λ) = max c i x i i=1 r s.t. a i x i λ, i=1 x i {0, 1}. If ˆx r = 0, then f r (λ) = f r 1 (λ), ˆx r = 1 then f r (λ) = c r + f r 1 (λ a r ). Thus we arrive at the recursion f r (λ) = max {f r 1 (λ), c r + f r 1 (λ a r )}. Martin Branda (KPMS MFF UK) 2017-03-20 34 / 39

Dynamic programming Dynamic programming 0. Start with f 1 (λ) = 0 for 0 λ < a 1 and f 1 (λ) = max{0, c 1 } for λ a 1. 1. Use the forward recursion f r (λ) = max {f r 1 (λ), c r + f r 1 (λ a r )}. to successively calculate f 2,..., f n for all λ {0, 1,..., b}; f n (b) is the optimal value. 2. Keep indicator p r (λ) = 0 if f r (λ) = f r 1 (λ), and p r (λ) = 1 otherwise. 3. Obtain the optimal solution by a backward recursion: if p n (b) = 0 then set ˆx n = 0 and continue with f n 1 (b), else (if p n (b) = 1) set ˆx n = 1 and continue with f n 1 (b a n )... Martin Branda (KPMS MFF UK) 2017-03-20 35 / 39

Dynamic programming Knapsack problem Values a 1 = 4, a 2 = 6, a 3 = 7, costs c 1 = 4, c 2 = 5, c 3 = 11, budget b = 10: max s.t. 3 c i x i i=1 3 a i x i 10, i=1 x i {0, 1}. Martin Branda (KPMS MFF UK) 2017-03-20 36 / 39

Dynamic programming Knapsack problem Dynamic programming a 1 = 4, a 2 = 6, a 3 = 7, c 1 = 4, c 2 = 5, c 3 = 11 r/ λ 0 1 2 3 4 5 6 7 8 9 10 1 0 0 0 0 4 4 4 4 4 4 4 f r 2 0 0 0 0 4 4 5 5 5 5 9 3 0 0 0 0 4 4 5 11 11 11 11 1 0 0 0 0 1 1 1 1 1 1 1 p r 2 0 0 0 0 0 0 1 1 1 1 1 3 0 0 0 0 0 0 0 1 1 1 1 Martin Branda (KPMS MFF UK) 2017-03-20 37 / 39

Dynamic programming Dynamic programming Other successful applications Uncapacitated lot-sizing problem Shortest path problem... Martin Branda (KPMS MFF UK) 2017-03-20 38 / 39

Literature Dynamic programming V. Klee, G.J. Minty, (1972). How good is the simplex algorithm?. In Shisha, Oved. Inequalities III (Proceedings of the Third Symposium on Inequalities held at the University of California, Los Angeles, Calif., September 1 9, 1969). New York-London: Academic Press, 159 175. G.L. Nemhauser, L.A. Wolsey (1989). Integer Programming. Chapter VI in Handbooks in OR & MS, Vol. 1, G.L. Nemhauser et al. Eds. L.A. Wolsey (1998). Integer Programming. Wiley, New York. L.A. Wolsey, G.L. Nemhauser (1999). Integer and Combinatorial Optimization. Wiley, New York. Northwestern University Open Text Book on Process Optimization, available online: https://optimization.mccormick.northwestern.edu [2017-03-19] Martin Branda (KPMS MFF UK) 2017-03-20 39 / 39