Optimizatio Methods MIT 2.098/6.255/15.093 Fial exam Date Give: December 19th, 2006 P1. [30 pts] Classify the followig statemets as true or false. All aswers must be well-justified, either through a short explaatio, or a couterexample. Uless stated otherwise, all LP problems are i stadard form. (a) If there is a uique primal optimal solutio to a liear programmig problem, the the reduced costs of all the obasic variables are strictly positive. (b) For a etwork flow problem with capacity costraits, there always exists a optimal solutio that is tree-structured. (c) Whe miimizig a covex fuctio over a covex set, the optimal solutio is always o the boudary of the set. (d) For a covex optimizatio problem with costraits, if a feasible poit satisfies the KKT coditios the it is a global optimum. (e) For a oliear optimizatio problem, if Newto s method coverges, the it coverges to a local miimum. (f) (g) For a LP i stadard form, if c 0 the the primal is either bouded or ifeasible. O very degeerate LP problems, the simplex method performs better tha iterior poit methods. (h) The primal iterates x k geerated by the affie scalig algorithm are always i the iterior of the primal feasible set. (i) The feasible set of a semidefiite programmig problem is always covex. (j) For a quadratic fuctio f(x) = x T Ax + b T x + c, the covergece rate of Newto s method depeds o the coditio umber of the matrix A. Solutio: (a) FALSE. This is true oly if the uique optimal solutio is odegeerate. (b) FALSE. I the capacitated case, optimal solutios do ot ecessarily have to be trees, a simple couterexample is a etwork with odes {A, B, C, D} ad edges A B, A C, B D, C D of uit capacity. (c) FALSE. Solutios ca be i the iterior. As a example, cosider miimizig x 2 o [ 1, 1]. (d) TRUE. This is prove i the book ad i the lecture otes. (e) FALSE. Newto s method ca coverge to global maxima. (f) TRUE. If c is oegative, the if the problem is feasible we have c T x 0, ad thus is it bouded. (g) FALSE. For degeerate problems, iterior poit methods are a much better choice. (h) TRUE. By costructio, the primal iterates i the affie scalig method (with β < 1) are always strictly feasible. (i) TRUE. The feasible set of a SDP problem is covex, sice it is the itersectio of two covex sets (a affie subspace ad the coe of PSD matrices). (j) FALSE. For quadratic fuctios, Newto s method coverges exactly i oe iteratio. 1
P2. [25 pts] You are plaig o havig access to a car for the ext N years, where N is a fixed umber. The price of a ew car is P dollars. For reliability reasos, you will oly ow relatively ew cars, at most m years old. The yearly cost of repairig ad maitaiig a car durig its kth year is r k, ad it satisfies r 1 < r 2 < < r m (i.e., it icreases over time). At the ed of ay give year, you have the optio of exchagig your k-year old car for a ew oe, with the correspodig trade-i value t k of your old car satisfyig t 1 > t 2 > > t m (i.e., depreciatig over time). You wat to fid the most ecoomical sequece of buys ad trade-is, i.e., wat to miimize the total cost over the N-year period. This icludes all the moey spet, either i buyig or repairig (otice that you ca sell your car at the ed of the N years). (a) Propose a shortest path (or etwork flow) formulatio for this problem. (b) Propose a dyamic programmig formulatio for this problem. Express clearly what are the state ad decisio variables, ad the correspodig iteratio. (c) Use your DP formulatio to solve the problem for the followig data: N = 5, m = 3, P = 20000, the repairig costs ad the trade-i values r 1 = 1200, r 2 = 1600, r 3 = 2400, t 1 = 16000, t 2 = 12000, t 3 = 10000. What is the optimal sequece of actios? Is it uique? Solutio: (a) Defie a etwork whose odes are o a N m grid with two additioal odes: a source ode s ad a sik ode t. A ode o the gride is idexed by (time idex) t = 1,..., N ad (car age) a = 1,..., m. From ode (t, a), there is a directed edge to ode (t + 1, a + 1) with cost r a if t < N ad a < m (car is maitaied for oe year) to ode (t + 1, 1) with cost P t k + r 1 if t < N (car is traded i) to a sik ode t if t = N with cost t a. Node s is coected to the grid ode (1, 1) with cost P + r 1. Node s a supply of 1 ad ode t a demad of 1. A mi-cost flow solutio will correspod to a optimal purchase/maiteace pla for the car over the time horizo N (b) Let the time idex t ru from 1 to N. Defie the state a at time t as the age of the car at the ed of the curret period. Let V t (a) be the expected cost give that the car is a-year old at the ed of the time period t. At time t < N, if a = m, the car has to be exchaged ad maitaied for oe year with cost P t m + r 1, whereas there are two available optios i state a {1,..., m 1}: maitai the car, with a cost of r a, leadig to (a + 1)-year old car at the ext period trade-i the car for a ew oe ad maitai it for a year, with a cost P t a + r 1, leadig to a oe-year old car i the ext period. At time t = N, the car is sold for its value t a. Bellma s equatios for this problem are V N (a) = t a, V t (m) = P t m + r 1 + V t+1 (1), V t (a) = mi (r a + V t+1 (a + 1), P t a + r 1 + V t+1 (1)), a = 1,..., m, t < N a = 1,..., m 1, t < N. 2
(c) Solvig Bellma s recursio yields the optimal cost of $26, 500. 3
P3. [20 pts] Cosider the followig trasshipmet problem: The supply odes are A, B, the demad odes are D, E, ad the trasshipmet ode is ode C. There are four ukows a, b, c ad d. The supply/demad amouts i the differet odes are: A : a, B : 400, D : b, E : 200, where as usual a positive amout idicates supply ad a egative amout idicates demad. We are iterested i fidig a optimal (miimum cost) trasshipmet pla. (a) State coditios o a, b, c, d such that the above problem is feasible. (b) Cosider the spaig tree give by the edges {(A, D), (B, C), (C, D), (C, E)}. State coditios o a, b, c, d for which the spaig tree solutio will be feasible. (c) State coditios o a, b, c, d for which the spaig tree solutio will be optimal. (d) State coditios o a, b, c, d for which there will be multiple solutios, icludig the spaig tree solutio idicated above. Solutio: (a) For the problem to be solvable, supply ad demad must balace, givig the ecessary coditio a + 400 = 200 + b, or equivaletly, b a = 200. (b) If the solutio has the structure of the idicated spaig tree, the the followig relatios must hold: a + 200 = b. Here we ca calculate all flows o this spaig tree ad the coditio agai is the flow balace coditio. (c) The optimality coditio for the spaig tree solutio is that all reduced costs of o-basic arcs are o-egative. I order to calculate the reduced costs, we eed to calculate ode potetials. Without loss of geerality, set p C = 0. Kowig the fact that c ij = c ij (p i p j ) = 0 for all basic arcs (i, j), we obtai p B = 5, p D = 3, p A = 1, ad p E = 2. The reduced costs for o-basic arcs the ca be calculated as follows: c BA = c 6, c CA = d 1, c BE = 2, c ED = 10 Thus i additio to the feasibility coditio b = a + 200, we obtai the optimality coditio for the idicated spaig tree solutio: { c 6 0. d 1 0 (d) We eed to have reduced costs of some o-basic arcs to be zero i order to have multiple optimal solutios, icludig this spaig tree solutio. It meas either c = 6 or d = 1 (i additio to the feasibility ad optimality coditios metioed previously i (c)). We ca see that there is a cost-equivalet path from B to D via A as compared to the path B C D i both cases. Thus, other optimal solutios ca be costructed by reroutig flows from B to D via A if { either c = 6 or d { = 1. The fial coditios for multiple optimal solutios are: a + 200 = b, ad c 6 = 0 c 6 0 or d 1 0 d 1 = 0. 4
P4. [25 pts] Cosider a set of poits {(x 1, y 1 ),..., (x, y )} i the plae. We wat to fid a poit (x, y) such that the sum of the Euclidea distaces from this poit to all the other poits is miimized. (a) Give a oliear optimizatio formulatio of this problem. (b) Is the objective fuctio differetiable? Is this a covex optimizatio problem? (c) Write the correspodig optimality coditios. Give a geometric iterpretatio of this coditio. (d) Are the optimality coditios ecessary? Sufficiet? State clearly your assumptios. (e) Provide a semidefiite programmig formulatio of this problem. All aswers ad explaatios must be fully justified. Solutio: (a) A simple formulatio as a ucostraied oliear optimizatio problem is the followig: mi (x x i ) 2 + (y y i ) 2. x,y (b) The objective fuctio is differetiable everywhere, except at the poits where (x, y) is equal to oe of the (x i, y i ). The objective fuctio is covex, sice it is a sum of covex fuctios. (c) The optimality coditios are obtaied by settig the gradiet equal to zero (we assume that the miimum occurs at a differetiable poit). f x x i = = 0 x (x, y) (x i, y i ) f y y i = = 0 y (x, y) (x i, y i ) This coditio ca be iterpreted as requirig the sum of the ormalized vectors from the poit (x, y) to the (x i, y i ) to be equal to zero. For istace, i the case = 2, the ay poit i the lie segmet betwee the two give poits will be optimal. Similarly, for = 3, the optimality coditio implies that the agles betwee the vectors from (x, y) to the other poits are all equal. (d) The optimality coditios are sufficiet, because the problem is covex. They are ecessary if the miimum occurs at a differetiable poit. (e) A semidefiite formulatio of the problem ca be easily obtaied if we itroduce slack variables d i ad formulate the problem as: mi d i s.t. (x x i ) 2 + (y y i ) 2 d 2 i i = 1,...,. The costraits are equivalet to: [ ] T [ ] [ ] d 2 x x i 1 0 x x i i 0 y y i 0 1 y y i Dividig by d i ad usig Schur complemets, we ca rewrite this as: d i x x i y y i mi d i s.t. x x i d i 0 0, i = 1,...,. which is a SDP formulatio. y y i 0 d i 5
1 MIT OpeCourseWare http://ocw.mit.edu 15.093J / 6.255J Optimizatio Methods Fall 2009 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. -