6.854J / J Advanced Algorithms Fall 2008

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1 MIT OpenCourseWare J / J Advanced Algorthms Fall 2008 For nformaton about ctng these materals or our Terms of Use, vst:

2 18.415/6.854 Advanced Algorthms October 27, 2008 Lecturer: Mchel X. Goemans Lecture 14 1 Introducton For ths lecture we ll look at usng nteror pont algorthms for solvng lnear programs, and more generally convex programs. Developng orgnally n 1984 by Narendra Karmarkar, there have been many varants (wth some of the keywords path followng, prmal-dual, potental reducton, etc.) on nteror pont algorthms, especally through the late 80s and early 90s. In the late 90s, people began to realze that nteror pont algorthms could also be used to solve semdefnte programs (or, even more generally, convex programs). As much as possble, we wll dscuss lnear programmng, semdefnte programmng, and even a larger class called conc programmng n a unfed way. 2 Lnear Programmng We wll start wth lnear programmng. Remember that n lnear programmng, we have: Prmal: Gven A R m n, c R n and b R m, fnd x R n : Mn c T x s.t. Ax = b, x 0. Its dual lnear program s: Dual: Fnd y R m : Max b T y s.t. A T y c. We can ntroduce non-negatve slack varables and rewrte ths as: Dual: Fnd y R m, s R n : Max b T y s.t. A T y + s = c, s 0. We know that, for a feasble soluton, x n the prmal, and a feasble soluton (y, s) n the dual, we know by complementary slackness that they wll both be optmal (for the prmal and the dual resp.) ff x T s = 0. Snce ths s the component-wse product of two non-negatve vectors, we can equvalently say: x j s j = 0 j. 2.1 Usng the Interor Pont Algorthm The nteror pont algorthm wll teratvely mantan a strctly feasble soluton n the prmal, such that for all values of j, x j > 0. Smlarly n the dual, t wll mantan a y and an s such that for all values of j, s j > 0. Because of ths strct nequalty, we can never reach our optmalty 14-1

3 condton stated above; however, we ll get very close, and once we do, we can show that a jump from ths non-optmal soluton (for ether the prmal or the dual) to a vertex of mproved cost (of the correspondng program) wll provde an optmal soluton to the (prmal or dual) program. In some lnear programs, t may not be possble to start wth a strctly postve soluton. For example, for any feasble soluton to the program, t may be that x j = 0, so we may be unable to fnd a strctly feasble soluton wth whch to start the algorthm. Ths can be dealt wth easly, but we wll not dscuss ths. We ll assume that the prmal and dual both have strctly feasble solutons. 3 Semdefnte Programmng As ntroduced n the prevous lecture, n semdefnte programmng, our varables are the entres of a symmetrc posttve semdefnte matrx X. Let S n denote the set of all real, symmetrc and n n matrces. For two such matrces A and B, we defne an nner product A B = A j B j = T race(a T B) = T race(ab). Semdefnte programmng (as a mnmzaton problem) s Mn j C X s.t. A X = b = 1...m X 0. Remember that for a symmetrc matrx M, M 0 means that M s postve semdefnte, meanng that all of ts (real) egenvalues λ 0, or equvalently, x, x T Mx Dual for SDP When workng wth lnear programs, we know the exstence of a dual lnear program wth a strong property: Any feasble dual soluton provdes a lower bound on the optmum prmal value and, f ether program s feasble, the optmum prmal and optmum dual values are equal. Does a smlar dual for a semdefnte progrm exst? The answer f yes, although we wll need some addtonal condton. We clam that the dual takes the followng form. Dual: Fnd y R n, and S S n : Max y R m s.t. b T y y A + S = C S

4 3.1.1 Weak Dualty For weak dualty, consder any feasble soluton x n the prmal, and any feasble soluton (y, S) n the dual. We have: ( ) C X = y A + S X = y (A X) + S X = y b + S X = b T y + S X b T y, the last nequalty followng from Lemma 1 below. Ths s true for any prmal and dual feasble solutons, and therefore we have z w, where: z = mn{c X : X feasble for prmal}, w = max{b T y : (y, S) feasble for dual}. Lemma 1 For any A, B 0, we have A B 0. Proof of Lemma 1: Any postve semdefnte matrx A admts a Cholesvky decomposton: A = V T V for some n n matrx V. Thus, A B = T race(ab) = T race(v T V B) = T race(v BV T ), the last nequalty followng from the fact that, for (not necessarly symmetrc) square matrces C and D, we have T race(cd) = T race(dc). But V BV T s postve defnte (snce x T V BV T x 0 for all x), and thus ts trace s nonnegatve, provng the result. A smlar lemma was used when we were talkng about lnear programmng, namely that f a, b R n wth a, b 0 then a T b Strong Dualty In general, t s not true that z = w. Several thngs can go wrong. In defnng z, we wrote: z = mn C X. However, that mn s not really a mn, but rather an nfmum. It mght happen that the nfmum value can be approached arbtrarly closely but no soluton may attan that value precsely. Smlarly n the dual, the supremum may not be attaned. In addton, n semdefnte programmng, t s possble that the prmal may have a fnte value, but that the dual may be nfeasble. In lnear programmng, ths was not the case. If the prmal had a fnte feasble value and was bounded, the dual was also fnte and wth the same value. In semdefnte programmng, the prmal can be fnte, whle the dual may be nfeasble or vce versa. In addton, both the prmal and dual could be fnte, but they could be of dfferng values. That all sad, n the typcal case, you do have strong dualty (z = w), but only necessarly under certan condtons Introducng a Regularty Condton Assume that the prmal and dual have a strctly feasble soluton. Ths means that for the prmal: X s.t. A X = b = (1...m). X

5 A 0 denotes that A s a postve-defnte matrx, meanng that a = 0, a T Xa > 0, or equvalently that all ts egenvalues λ satsfy λ > 0. Lkewse, n the dual, there exsts y and S such that: y A + S = C S 0. If we assume ths regularty condton that we ve defned above, then the prmal value z s fnte and attanable (.e. t s not an nfmum, but actually a mnmum), and the dual value w s attaned and furthermore z = w. Ths s gven wthout proof. 4 Conc Programmng Conc Programmng s a generalzaton of both Lnear Programmng and Semdefnte Programmng. Frst, we need the defnton of a cone: Defnton 1 A cone s a subset C of R n that has the property that for any v C and λ R +, λv s also n C. Conc Programmng s constraned optmzaton over K, a closed convex cone, wth a gven nner product x, y. We can, for example, take K = R n and x, y = x T y for any x, y R n ; ths wll lead to lnear programmng. Conc programmng, lke LP and SDP, has both a prmal and a dual form; the prmal s: Prmal: Gven A R m n, b R m, and c R n : mn c, x s.t. Ax = b x K. More generally, we could vew K as a cone n any space, and then A s a lnear operator from K to R m. To form the dual of a conc program, we frst need to fnd the polar cone, K, of K. The polar cone s defned to be the set of all s such that for all x n K, s, x 0. For nstance, the polar cone of R n + s R+ n tself (ndeed f s j < 0 then we have s / K snce e j, s < 0; conversely, f s 0 then x, s 0). In the case that K = K, we say that K s self-polar. Smlarly, the polar cone of P SD, the set of postve semdefnte matrces, s also tself. We also defne the adjont (operator) A of A to be such that, for all x and y, A y, x = y, Ax. For example, f the nner product s a standard dot product and A s the matrx correspondng to a lnear transformaton from R n to R m, then A = A T. To wrte the conc dual, we ntroduce a varable y R m and s R n and optmze: Dual: Weak Dualty max b, y s.t. A y + s = c s K. We can prove weak dualty that the value of the prmal s at least the value of the dual as follows. Let x be any prmal feasble soluton and (y, s) be any dual feasble soluton. Then c, x = A y + s, x = A y, x + s, x = y, Ax + s, x = b, y + s, x b, y, where we have used the defnton of K to show that s, x 0. Ths means that z, the nfmum value of the prmal, s at least the supremum value w of the dual. 14-4

6 4.0.5 Strong Dualty In the general case, we don t know that the two values wll be equal. But we have the followng statement (analogous to the regularty condton for SDP): f there exsts an x n the nteror of K, such that Ax = b, and a s n the nteror of K, wth A y + s = c, then the prmal and the dual both obtan ther optmal values, and those values are equal. 4.1 Semdefnte Programmng as a Specal Case of Conc Programmng LP s a specal case of conc programmng, f we let K = R n and take the nner product to be the + standard dot product a, b = a T b. We can also make any SDP nto a conc program; frst, we need a way of transformng semdefnte matrces nto vectors. Snce we are optmzng over symmetrc matrces, we ntroduce a map svec(m) that only takes the lower trangle of the matrx (ncludng the dagonal). To be able to use the standard dot product wth these vectors, svec multples all of the off-dagonal matrces by 2. So svec maps X to As a result: n svec(x), svec(y ) = x y + (x 11, x 22,..., x nn, 2x 12 2x13,..., 2x (n 1)n ). 2xj 2yj = Ths means that usng the basc dot product as the nner product s compatble wth the nner product used n SDP. So we can formulate an SDP as a conc program by lettng K = {svec(x) : X 0}, whch s a closed convex cone. To show convexty, we need to show that f A and B are matrces n P SD, then λa + (1 λ)b s also n P SD for 0 λ 1. Indeed, for any vector v, we have ( ) ( ) v T (λa + (1 λ)b)v = λ v T Av + (1 λ) v T Bv 0. m A (y), X := y, A(X) = y A X, m mplyng that A maps y to =1 y A. The dual SDP now arses as the dual conc program. =1 1 <j n 1,j n x j y j = T r(ab) = A B. Then, we can let the matrx A be a matrx that s the composton of the correspondng A of the semdefnte program, so that A svec(x) = (A X) =1,...,m. Now that the semdefnte program s cast nto a conc program, we could wrte the conc dual, and one could verfy that what we get s precsely the dual of the semdefnte program we defned ( ) earler. Instead of mappng the space of symmetrc matrces (say p p) nto R n (wth n = p+1 2 ) usng svec( ), one could smply defne K = {X S p : X 0} and X, Y = X Y. Now our lnear operator A : S n R m then maps X nto (A X) =1,,m. Its adjont A : R m S n s defned by: 4.2 Barrer Functons To solve the conc program, we wll requre a barrer functon F. Ths s a functon from nt(k), the nteror of K, to R such that 1. F s strctly convex, 2. F (x ) as x x K, where K s the boundary of K. =1 14-5

7 We wll use the barrer functon to punsh canddate solutons that are close to the boundary of K, keepng the current pont nsde K. Good barrer functons, that result n a fast overall n algorthm, have more propertes that wll be descrbed n a later lecture. For K = R +, a good barrer functon s F (x) = log(x ). As any one of the coordnates approaches 0, the log approaches, so the total functon goes to. One can also check that ths functon s strctly convex. For K = svec(p SD p ) or more smply K = P SD p (the set of symmetrc p p postve semdefnte matrces), the nteror of K s the set of postve defnte matrces, whch all have strctly postve determnants. (Ths s because the determnant s equal to the product of the egenvalues, whch are all strctly postve for a postve defnte matrx.) So we can use the followng barrer functon: F (X) = log(det(x)). As X approaches the boundary of K, the determnant goes to zero, and F goes to nfnty. One can also check that ths functon s strctly convex (ts Hessan, the matrx of second dervatves, can be shown to be postve defnte). 4.3 A Prmal-Dual Interor-Pont Method Once we have a barrer functon, we wll set the objectve functon of the prmal to c, x + µf (x), where µ s a parameter that we wll adjust through the course of the algorthm. Assumng that we start wth an ntal canddate that belongs to nt(k), we can gnore the constrant that x K, snce that wll be enforced through the barrer functon, snce there wll be an nfnte penalty for leavng K. Our prmal barrer problem BP (µ) wll be: mn{ c, x + µf (x) : Ax = b}. Analogously, for the dual, we change the objectve functon to b, y µf (s), where F s a barrer functon for the dual; we can also elmnate the constrant that s K. Our dual barrer problem, BD(µ), s: max{ b, y µf (s) : A y + s = c}. The basc method of the algorthm s to have a current value of µ, and keep track of the optmal solutons n the prmal BP (µ) and dual BD(µ). As long as µ s not zero, there s a unque optmum soluton for both, snce the objectve functon s the sum of a lnear functon and a strctly-convex functon, whch results n a strctly-convex functon. We wll steadly decrease µ, and keep track of the optmal solutons as they change; the paths the optmum solutons trace out s called the central path (or central trajectory). We wll show that the (prmal and dual) central paths wll converge to an optmum value of the prmal and dual orgnal programs. In the specal case of lnear programmng, once we are suffcently close, we can round the current soluton to the nearest vertex to obtan an optmum soluton. For semdefnte programmng, though, we do not have such an algorthm to convert a soluton for small enough µ to an optmum soluton. Let s characterze the optmum soluton to BP (µ) and BD(µ). We derve now the so-called KKT optmalty condtons. If there were no constrants n the conc program, then the mnmum would be found when the gradent of the objectve functon s zero. If there are affne constrants lke Ax = b, however, the mnmum wll occur when the gradent s normal to the affne space of feasble solutons. Otherwse, we could move along the projecton of the gradent on the feasble space, and mprove our objectve functon. For smplcty, let s frst look at the case when K = K n = R +, and the barrer functon s F (x) = log(x ). The objectve functon of the prmal s c, x µf (x), and the partal dervatves are µ ( c, x µf (x)) = c j x j x j 14-6

8 so the gradent s c µx 1, where x 1 denotes the vector {1/x }. But snce ths gradent s normal to the constrant Ax, the gradent must be of the form A T y for some y. So f we let s = µx 1, then we know c s s of the form A T y, or equvalently, The last constrant s equvalent to A T y + s = c s = µx 1. x j s j = µ (1) for all j. Now, lookng at the dual: the gradent wth respect to y s b, whch must be of the form Ax for some x. The gradent wth respect to s s µs 1, whch must equal the same x. Ths means that Ax = b s = µx 1, and the last equalty s agan equvalent to (1). So f we denote by x(µ) the optmum soluton to the prmal BP (µ) and by (y(µ), s(µ)) the optmum soluton to the dual BD(µ), one observes that each of them s a certfcate of optmalty for the other and furthermore: x j (µ)s j (µ) = µ. Ths means that the dualty gap n the orgnal prmal/dual par of lnear programs s x T s = nµ and therefore the dualty gap goes to 0 as µ goes to 0. Thus the central path (x(µ), y(µ), s(µ)) wll converge to optmum solutons to both the prmal and dual lnear programs. 14-7