1 15.093J/2.098J Optimization Methods Assignment 6 Solutions Exercise 6.1 These functions are twice dierentiable; therefore, we can check the convexity using the Hessian matrices. A function f (x) is convex if its Hessian matrix is positive semidenite for
1 15.093J/2.098J Optimization Methods Assignment 5 Solutions Exercise 5.1 BT, Exercise 10.2.
The decision variables are xi , i = 1, 20. xi = 1 if the player pi is selected; otherwise, pi = 0. We have:
The total number of players in the team is 12: 20
1 15.093J/2.098J Optimization Methods Assignment 4 Solutions Exercise 4.1 BT, Exercise 7.1. Construct the network as follows. For each day i, create two nodes: node di for dirty tablecloths and ci for clean tablecloths. There is a supply of ri for node di
Math464 - HW 7 Due on Friday, Mar 5
1
Linear Optimization (Spring 2010) Brief solutions to Homework 7
1. We start the standard 12 A = 2 1 22 form LP with 2100 20 2 0 1 0 , b = 20 , and cT = 10 12 12 0 0 0 . 1001 20
The details of the revised simplex metho
Math464 - HW 6 Due on Thursday, Feb 25
1
Linear Optimization (Spring 2010) Brief solutions to Homework 6
1. To get a contradiction, assume that there is a point x S with f (x) < f (x ). Notice that x has to be non-local to x , i.e., x x > , else it will v
MIT 2.098/6.255/15.093 Optimization Methods Prof. J. Vera, Fall 2007 Homework Assignment 5. Solution Problem 1: BT Ex 7.1 Solution: (a) Construct the network as follows: For each day i, create two nodes: Node ci for clean tableclths with supply ri , Node
Math464 - HW 5 Due on Thursday, Feb 18
1
Linear Optimization (Spring 2010) Brief solutions to Homework 5
1. As seen in class (in Lecture 11), there may be some change of variables when we convert a polyhedron in general form to standard form. In particula
Math464 - HW 4 Due on Friday, Feb 12
1
Linear Optimization (Spring 2010) Brief solutions to Homework 4
1. Procedure for nding basic solutions: As in the original case, we still start by choosing the indices for m linearly independent columns, B (1), . . .
Math464 - HW 3 Due on Thursday, Feb 4
1
Linear Optimization (Spring 2010) Brief solutions to Homework 3
1. We rewrite the second constraint for cash availability in the LP formulation (given in the solutions to Homework 1) as follows: max 3x1 + 3.4x2 s.t.
MIT 2.098/6.255/15.093 Optimization Methods Prof. J. Vera, Fall 2007 Homework Assignment 2. Solution Problem 1: BT, exercise 3.2 Solution. (a) If feasible solution x is optimal, suppose there exists a feasible direction d such that c d < 0, then for a sma
Math464 - HW 2 Solutions Due on Thursday, Jan 28
1
Linear Optimization (Spring 2010) Brief solutions to Homework 2
1. We rst prove that f is convex when each fi is convex. (a) Since the fi s are convex, [0, 1], and for x, y Rn , we have fi ( x + (1 )y ) f
Math464 - HW 1 Solutions Due on Thursday, Jan 21
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Linear Optimization (Spring 2010) Brief solutions to Homework 1
1. The standard form LP is the following. min s.t. 3x1 + (x+ x ) 3 3 x2 4(x+ x ) e1 =4 3 3 + + s2 = 2 x1 5x2 (x3 x3 ) x1 , x2 , x+ , x , e1
MIT 2.098/6.255/15.093 Optimization Methods Prof. J. Vera, Fall 2007 Homework Assignment 1. Solution Problem 1: BT, exercise 1.4 Solution: One way to reformulate it as a linear programming problem is as follows:
minimize 2x1 + 3z subject to x1 + 2 + x2 5
MIT 2.098/6.255/15.093 Optimization Methods Prof. J. Vera, Fall 2007 Homework Assignment 4. Solution Guidelines The purpose of this assignment is to give you the opportunity to solve a real problem using available software, and to improve the solution pro
EE292 Analysis and Control of Markov Chains Prof. Ben Van Roy
Spring 2008 April 2, 2006
Homework Assignment 1: Due April 10 Deterministic DP Formulations
Problems 2.2 and 1.16 from the text.
Viterbi Deco ding
Consider the trellis code of Figure 2.2.5 of t
EE292 Analysis & Control of Markov Chains Prof. Ben Van Roy
Spring 2006 June 9 , 2006
Homework Assignment 8 : Solutions
1 a) We consider each customer as a bandit so that we have
i
xi bandits in all. The state of a bandit
is simply the queue the bandit be
EE292 Analysis & Control of Markov Chains Prof. Ben Van Roy
Spring 2006 June 7 , 2006
Homework Assignment 7 : Solutions
1 We use the notation s for our state variable; x for queue length, i to indicate on/off. S = cfw_(x, i) : x cfw_0, 1, . . . , 100, i c
EE292 Analysis & Control of Markov Chains Prof. Ben Van Roy
Spring 2006 May 29 , 2006
Homework Assignment 6 : Solutions
1 The optimal policy is to keep the server on if it is on, and to turn the server on if it is off. See the attached code. Starting at t
EE292 Analysis & Control of Markov Chains Prof. Ben Van Roy
Spring 2006 May 21, 2006
Homework Assignment 5 : Solutions
5.2 a) We have a linear quadratic problem with imperfect state information, where Ak , Bb , Qk , and Rk are all 1. Consequently, the opt
EE292 Analysis & Control of Markov Chains Prof. Ben Van Roy
Spring 2006 May 10, 2006
Homework Assignment 4 : Solutions
4.13 Let xk be the gamblers fortune at time k. Let k = 1 if the gambler loses at time k and 1 if he wins at time k, where P r(k = 1) = 1
EE292 Analysis & Control of Markov Chains Prof. Ben Van Roy
Spring 2006 April 29, 2006
Homework Assignment 3 : Solutions
1 See attached code. Note that we use the fact that argmaxp0 exp(-p)(p + C) = 1/ - C for C > 0. Below is a plot of optimal price versu
EE292 Analysis & Control of Markov Chains Prof. Ben Van Roy
Spring 2006 April 21, 2006
Homework Assignment 2 : Solutions
2 The system is a linear system of the type discussed in class. In particular, we have the system equation: xk+1 = Axk + Buk + k where
EE292E Analysis & Control of Markov Chains Prof. Ben Van Roy
Spring 2008 April 10, 2008
Homework Assignment 1 : Solutions
1.16 (a) Each state is a set Sk cfw_2, .N . The allowable states at stage k are those sets Sk of cardinality k . The allowable contro
EE 292E Analysis and Control of Markov Chains Prof. Ben Van Roy
Spring 2008 May 29, 2008
Homework Assignment 8: Due June 2 Priority Assignment
Consider a variation of the queueing priority assignment problem in the last homework. In particular, there are
EE 292E Analysis and Control of Markov Chains Prof. Ben Van Roy
Spring 2008 May 22, 2008
Homework Assignment 7: Due May 29 Queueing with Constraints
Consider the Queueing problem from Homework 6, but suppose that the objective is to minimize average cost,
EE 292E Analysis and Control of Markov Chains Prof. Ben Van Roy
Spring 2008 May 15, 2008
Homework Assignment 6: Due May 22 Queuing
Consider a queue with a nite buer of length of 100. At the beginning of each time period, a customer arrives with probabilit
EE 292E Analysis and Control of Markov Chains Prof. Ben Van Roy
Spring 2008 May 8, 2008
Homework Assignment 5: Due May 15 LQG
Do problem 5.2 from the textbook.
Slots
Do problem 5.9 from the textbook.
Computer Manufacturer
Do problem 7.3 from the textbook.
EE 292E Analysis and Control of Markov Chains Prof. Ben Van Roy
Spring 2008 April 24, 2008
Homework Assignment 4: Due May 1 Gambling
Do problem 4.13 from the textbook. Also answer the question for the case where the objective is to maximize: E [x1a /(1 a)