Department of Mathematics Oleg Burdakov of 30 October Consider the following linear programming problem (LP):

Transcription

1 Linköping University Optimization TAOP3(0) Department of Mathematics Examination Oleg Burdakov of 30 October 03 Assignment Consider the following linear programming problem (LP): max z = x + x s.t. x x x + x + x 3 = x, x, x 3 0 (p) a) Solve the problem with the simplex method starting from the point for which x and x are nonbasic variables. What are the optimal values of x,x,x 3 and z? Is the optimal basic feasible solution degenerate? (p) b) Is the optimal solution obtained in a) unique? If not, find ALL alternative optimal solutions. (p) c) Formulate the LP-dual to the problem (LP). (p) d) Use the complementarity conditions to check if the point x = (3, 0, ) T is optimal. (p) e) How the optimal value of z would change (increase? decrease? remain the same?) if the right-hand side of the first constraint was increased a little? (Apply the sensitivity analysis) (p) f) Suppose a new variable x 4 is introduced in (LP) with the objective function coefficient c 4 = and constraint column a 4 = ( 3, ) T. Will the objective function value increase, decrease or remain the same? (Apply the sensitivity analysis.) Assignment (p) a) For some integer programming problem, its LP relaxation gives the following optimal simplex tableau bas. z x x x 3 x 4 r.h.s. z x x Is this solution to LP relaxation optimal to the original integer programming problem? Find a Gomory cut for this solution. (p) b) Solve the following integer programming problem using Branch & Bound. max x +x s.t. x +x 6 6x +x 30 x,x 0 and integer

2 LP relaxation may be solved graphically. If the both components are fractional, branch on x. Choose the branch first. Follow the last-in-first-out rule. Assignment 3 Consider the function f(x) = x x x +x 4 +x 3 +x. (p) a) Determine whether d = (0, ) T is a descent direction for the point x = (7, ) T. (p) b) Consider the problem of minimizing f(x). Make one iteration of the steepest descent method starting from the same point x = (7, ) T. (3p) c) Find all stationary points of this function, and determine whether they are local minimizers and maximizers. Assignment 4 Consider the following nonlinear programming problem (P) min x +x s.t. (x ) +x 4 x,x 0 (p) a) Is (P) a convex programming problem? (p) b) Solve (P) graphically. Find all local and global minima. (p) c) Write the Karush-Kuhn-Tucker (KKT) conditions for the given problem (P). The conditions must be written for the specific functions that define (P). (p) d) Check if the KKT conditions hold at the point x = (0 0) T. (p) e) Find a KKT point for which x > 0 and x > 0. Show algebraically (not geometrically) that the KKT conditions hold for this point. (p) f) The penalty method reduces(p) to an unconstrained minimization problem min F(x, c), where c is a penalty parameter. Compute F(x,) at the point x = ( ) T.

3 Assignment There are five sub-networks of the form where i =,,3,4,. In each of them, it is possible to send a flow from its left side to its right side (l r), and from its right side to its left side (r l). The maximal l r and r l flows are given for each sub-network N i by the table: N i l r r l N 3 N 3 N 3 4 N 4 N 3 Consider the following network composed of the given five sub-networks and four nodes Each edge connecting nodes and sub-networks has unlimited capacity for sending flow in any direction. Consider the problem of sending maximum flow f from node to node 4. (p) a) Reformulate this problem as a standard maximum flow problem (construct a new directed graph, indicate upper and low bounds for each edge). (3p) b) Use the max-flow algorithm for finding the maximum flow f in the new directed graph. (p) c) Find a minimum cut in the new directed graph. 3

4 Assignment 6 Consider the network (p) a) Find the shortest path from node to node. (p) b) What is the optimal solution to the linear programming formulation of the shortest path problem solved in a)? (4p) c) An owner of a network of hotels located in seven cities wishes to arrange a quick check of the air-conditioning systems in all his hotels. He is going to send workers to do this job. They start in city s. To travel from one city to another, they use hotel s cars (say, with unlimited number of sits). In the graph given below, each arc (i,j) is labeled with the cost associated with one-car traveling between city i and city j s The owner does not pay for their return trip (for instance, because each of the workers is offered, as a bonus, to spend a vacation in the last hotel that this worker will check, but they must pay themselves for returning back after the vacation). The owner is interested in minimizing the total traveling cost that he pays. Use a proper algorithm for finding the minimal cost. What is the optimal traveling strategy? (For each city, excluding the terminal ones, indicate the city or the cities to go next and the number of workers in each car.) What is the minimal number of workers that ensures the minimal traveling cost? Assignment 7 ( p.) Use the dynamic programming to solve the problem max 3x +4x +6x 3 s.t. 4x +x +7x 3 9 x,x,x 3 {0,} 4

5 ANSWERS Assignment a) The simplex method starts in x (0) = (0, 0,) T and generates the points x () = (0,,3) T, x () = (,,0) T. The point x () is optimal with z =. This optimal solution is not degenerate, because all the basic variables are strictly positive. b) The optimal solution is not unique. The alternative solution is x (3) = (, 0,0) T. All optimal solutions: x(λ) = λ +( λ) 0, 0 λ. 0 0 c) The dual problem: min w = v + v s.t. v + v v + v v 0 v 0, v is free d) The complementary slackness: () v (x x +) = 0 () x (v +v ) = 0 (3) x ( v +v ) = 0 (4) x 3 v = 0 For x = (3, 0, ) T, this implies: () v = 0 () v +v = 0 (4) v = 0 Obviously, there are no values of v and v which would satisfy these conditions. Thus, the complementary slackness is not satisfied for any v and v. This means that the point x = (3, 0, ) T is not optimal. e) The value of z would not change because the dual price of the st constraint is equal to zero (see the optimal simplex tableau). f) For any of the two optimal solutions x () or x (3), the complementary slackness gives the optimal values of the dual variables: v = 0, v =. Then c 4 = c 4 v Ta 4 = ( 0 )( ) 3 = 3. Since c 4 < 0, the optimal value z will not be improved (it will remain the same). Assignment a) The solution to the LP relaxation problem is not optimal to the original integer programming problem, because the value of x is not integer. The corresponding row of the simplex tableau gives the following equivalent Gomory cuts: x x 3 +x 4 0 and 0. 0.x 3 0.x 4 0.

6 b) P0: x LP = (7/, 9/) T, z LP =.3 UBD =, which is an upper bound for the original problem. P = P0 & (x 3): x LP = (3, ) T, z LP =. Do not continue branching further from this node, because x LP is integer. LBD =. Stop, because LBD = UBD =, i.e. it is impossible to improve the currently available candidate solution. The optimal solution: x = (3, ) T, z =. Assignment 3 a) d = (0, ) T is not a descent direction for the point x = (7, ) T, because d T f( x) = 0. b) x = x 0 +t 0 d 0 = (, ) T, where x 0 = (7, ) T, d 0 = (, 0) T and t 0 = /. c) The function has three stationary points: x = (0, 0) T, x = ( 0., 0.) T and x 3 = (, ) T. The points x and x 3 are local minimizers. The point x is neither local minimizer, nor local maximizer. Assignment 4 a) It is not a convex programming problem, because the feasible set is not convex. To show this we consider the two points a = (0, 0) T and b = (4, 0) T. These points are feasible, but λa+( λ)b is infeasible for λ = / (the st constraint is violated). b) Point a is a global minimizer, point b is a local minimizer. c) The ( KKT ) conditions: ( ) ( ) ( ) (x ) 0 +λ +λ +λ 0 3 = 0, λ 0, λ 0, λ 3 0, λ ((x ) +x 4) = 0, λ ( x ) = 0, λ 3 ( x ) = 0, (x ) +x 4, x 0, x 0. d) By the definition, x is a KKT point if there exist values of the Lagrangean multipliers such that the KKT conditions hold. Such values of the multipliers may not be unique when the gradients of the active constraints are linearly dependent. This is the case for x = (0, 0) T. It is a KKT point, because the KKT conditions hold, for instance, for λ = 0, λ = and λ 3 =. e) (3, 3) T has non-negative components and it is a KKT point. It is possible to find it, either geometrically, or from the KKT conditions. The requirement x > 0 and x > 0 implies λ = λ 3 = 0. The first of the KKT conditions give λ = and x = 3. Since λ 0, the first of the complementarity conditions implies (x ) +x = 4. From this equation, we obtain x = 3. f) For x = ( ) T, F(x,) = 8. 6

7 Assignment a) The problem can be reformulated with the use of the graph presented above. The arcs are labeled with upper bounds and optimal flow obtained in b). All low bounds are zero. It is required to find the maximal flow f from node to 4. b) The initial flow is zero. It requires to make two iterations of the max-flow algorithm to solve the problem. At the first iteration, modified Dijkstra s algorithm gives the path 3 4 with capacity 3. At the second iteration, modified Dijkstra s algorithm gives the path 3 4 with capacity. Thus, the maxflow f =. c) N s = {} is a min-cut. Assignment 6 a) Dijkstra s algorithm is not applicable here, because there is a negative arc cost. Notice that the nodes are topologically ordered. Then it is possible to apply Bellman s equations recursively. This gives y = 0, y = 6, y 3 =, y 4 = 3, y = 4 and the corresponding list of predecessors. Thus the shortest path is 3 4, and its length is 4. b) x = x 3 = x 34 = x 4 = and x ij = 0 for the other arcs. c) This is a minimum spanning tree problem. The minimal cost is 8. The optimal traveling strategy is the following. Two workers use one car on the path s 3. Then one of them uses one car on the path 6, and the other one uses one car on the path 4 7. The minimal number of workers is two. Assignment 7 The optimal solution is x = (,, 0), z = 7. 7