Robust Soutions of Uncertain Linear Programs A. Ben-Ta and A. Nemirovski 1) Abstract We treat in this paper Linear Programming (LP) probems with uncertain data. The focus is on uncertainty associated with hard constraints: those which must be satisfied, whatever is the actua reaization of the data (within a prescribed uncertainty set). We suggest a modeing methodoogy whereas an uncertain LP is repaced by its Robust Counterpart (RC). We then deveop the anaytica and computationa optimization toos to obtain robust soutions of an uncertain LP probem via soving the corresponding expicity stated convex RC program. In particuar, it is shown that the RC of an LP with eipsoida uncertainty set is computationay tractabe, since it eads to a conic quadratic program, which can be soved in poynomia time. Keywords: inear programming, data uncertainty, robustness, convex programming, interior-point methods 1 Introduction The data A, b associated with a inear program { } min c T x Ax b (1) are uncertain to some degree in most rea word probems 2). In many modes the uncertainty is ignored atogether, and a representative nomina vaue of the data is used (e.g., expected vaues). The cassica approach in Operations Research/Management Science to dea with uncertainty is Stochastic Programming (SP) (see, e.g., [6, 13] and references therein). But even in this approach constraints may be vioated, with certain penaty (this is the case for SP with recourse [6, 9], Scenario optimization [14], Entropic Penaty methods [1]) or with certain probabiity (chance constraints). In the dominating penaty approach, even when the random variabes are degenerate (deterministic), the corresponding SP mode does not recover necessariy the origina LP constraints, but ony a reaxation of these constraints. Thus, athough this was not stated expicity in the past, SP treats in fact mainy soft constraints. These remarks appy aso to the recent scenario-based penaty approach of Muvey, Vanderbei and Zenios [11]. In this paper we study uncertainty associated with hard constraints, i.e., those which must be satisfied whatever is the reaization of the data (A, b) within, of course, a reasonabe prescribed uncertainty set U. Feasibiity of a vector x is thus interpreted as Ax b (A, b) U. (2) Consequenty, we ca the probem min { c T x Ax b } (A, b) U (3) 1 The research was party supported by The Fund for the Promotion of Research at the Technion and by the Israe Ministry of Science grant # 9636-1-96 { 2) If the vector c is aso uncertain, we coud ook at the equivaent formuation of (1): min x,t t : c T x t, Ax b } and thus without oss of generaity we may restrict the uncertainty to the constraints ony. 1
the robust counterpart of the uncertain LP probem (1), and we ca a vector x soving (3) a robust soution of the uncertain probem. No underying stochastic mode of the data is assumed to be known (or even to exist), athough such knowedge may be of use to obtain reasonabe uncertainty sets (see Section 4). Deaing with uncertain hard constraints is perhaps a novety in Mathematica Programming; to the best of our knowedge, the ony previous exampe reated to this question is due to A.L. Soyster [16] (see beow). The issue of hard uncertain constraints, however, is not a novety for Contro Theory, where it is a we-studied subject forming the area of Robust Contro (see, e.g., [17] and references therein). The approach refected by (3) may ook at a first gance too pessimistic : the point is that there are indeed situations in reaity when an appicabe soution must be feasibe for a reaizations of the data, and even a sma vioation of the constraints cannot be toerated. We have discussed such a situation esewhere, for probems of designing engineering structures (e.g., bridges), see [2]. In these probems, ignoring even sma changes in the forces acting on the structure may cause vioent dispacements and resut in a severey unstabe structure. This exampe is not from the word of LP, but we can easiy imagine simiar situations in LP modes. Consider, e.g., a chemica pant which takes raw materias, decompose them into components and then recombine the components to get the fina product (there coud be severa decomposition recombination stages in the production process). The corresponding LP mode incudes the inequaity constraints expressing the fact that when recombining the intermediate components, you cannot use more than what is given by the preceding decomposition phase, and these constraints indeed are hard. If, as it is normay the case, the content of the components in raw materias is uncertain or/and the yied of the decomposition process depends on uncontroed factors, we end up with an LP program with uncertain data in hard constraints. As it was aready mentioned, uncertain hard constraints in LP modes were discussed (in a very specific setting) by A.L. Soyster [16] (for further deveopments, see aso [15, 7]). The case considered in these papers is the one of coumn-wise uncertainty, i.e., the coumns a i of the constraint matrix in the constraints Ax b, x 0 are known to beong to a given convex sets K i, and the inear objective is minimized over those x for which n x i a i b (a i K i, i = 1,..., n). (4) As it is easiy shown in [16], the constraints (4) are equivaent to a system of inear constraints A x b, x 0, a ij = sup a i K i (a i ) j. (5) It turns out that the phenomenon that (4) is equivaent to a inear system (5) is specific for coumnwise uncertainty. As we sha see ater, the genera case is the one of row-wise uncertainty (the one where the rows of the constraint matrix are known to beong to given convex sets). In this atter case, the robust counterpart of the probem is typicay not an LP program. E.g., when the uncertainty sets for the rows of A are eipsoids, the robust counterpart turns out to be a conic quadratic probem. Note aso that the case of coumn-wise uncertainty is extremey conservative: the constraints (5) of the robust counterpart correspond to the case when every entry in the constraint matrix is as bad (as arge) as it coud be. At the same time, in the case of row-wise uncertainty the robust counterpart is capabe to refect the fact that genericay the coefficients of the constraints cannot be simutaneousy at their worst vaues. The rest of the paper is organized as foows. Section 2 contains forma definition of the robust counterpart of an uncertain Linear Programming program (which is the one just defined, but in a sighty more convenient form). Here we aso answer some natura questions about the inks between 2
this counterpart and the usua LP program corresponding to the worst reaization of the data from the uncertainty set U. Note that the robust counterpart probem (P U ) has a continuum of constraints, i.e., is a semi-infinite probem, and as such it ooks computationay intractabe. In the ast part of Section 2 we discuss the crucia issue of which geometries of the uncertainty set U resut in a computationay tractabe robust counterpart of (1). Since we have in mind arge scae appications of robust LP s, we focus on geometries of the uncertainty set eading to expicit robust counterpart of nice anaytica structure, and which can be soved by high-performance optimization agorithms (ike the interior point methods). Specificay, in Section 3 we consider the case of our preferred structure where U is an intersection of finitey many eipsoids and derive for this case the expicit form of the robust counterpart of (1), which turns out to be a conic quadratic probem. Section 4 contains a simpe portfoio seection exampe iustrating our robust soution and comparing it to the soution obtained by the scenario-based approach of [11]. 2 Robust counterpart of an uncertain Linear Programming program 2.1 The construction For convenience and notationa simpicity we choose to cast the LP probem in the form (P ) min{c T x Ax 0, f T x = 1}. (6) The canonica LP program (1) in the Introduction can be obtained from (6) by the correspondence c := ( c 0 ), A := [A; b], f = (0,..., 0, 1) T. In the formuation (6), the vectors c, f R n are fixed data, and the uncertainty is associated ony with the m n matrix A. The uncertainty set, of which A is a member, is a set U of m n rea matrices. The robust counterpart of (6) is defined to be the optimization probem (P U ) min{c T x Ax 0 A U; f T x = 1}. (7) Thus, a robust feasibe (r-feasibe for short) soution to the robust counterpart of (P ) shoud, by definition, satisfy a reaizations of the constraints from the uncertainty set U, and a robust optima (r-optima for short) soution to (P U ) is an r-feasibe soution with the best possibe vaue of the objective. In the rest of this section we investigate the basic mathematica properties of program (P U ), with emphasis on the crucia issue of its computationa tractabiity. From now on we fix certain LP data U, c, f and denote by P the corresponding uncertain LP program, i.e., the famiy of a LP programs (P) with given c, f and some A U. Each program of this type wi be caed an instance (or a reaization) of the uncertain LP program. Evidenty, the robust counterpart (P U ) of an uncertain LP program remains unchanged when we repace the uncertainty set U by its cosed convex hu. Consequenty, from now on we aways assume that U is convex and cosed. 2.2 Robust counterpart and the worst LP program from P The first issue concerning the robust counterpart of an uncertain LP is how conservative it is. From the very beginning our approach is worst case oriented ; but coud (P U ) be even worse than the worst instance from P? E.g., (A) Assume (P U ) is infeasibe. Does it mean that there is an infeasibe instance (P ) P? 3
Further, assume that (P U ) is feasibe, and its optima vaue c is finite. By construction, c is greater than or equa to the optima vaue c (P ) of every probem instance (P ) P. The question of how conservative is the robust counterpart now may be posed as (B) Assume that (P U ) is feasibe with finite optima vaue c. Does it mean that there exists a probem instance (P ) P with c (P ) = c? A gap between the sovabiity properties of the instances of an uncertain LP and those of its robust counterpart may indeed exist, as is demonstrated by the foowing simpe exampe: a 11 x 1 + x 2 1 x min x 1 + x 2 s.t. 1 + a 22 x 2 1 x 1 + x 2 = 1 x 1, x 2 0 The uncertain eements are a 11 and a 22, and the uncertainty set is U = {a 11 + a 22 = 2, 1 2 a 11 3 2 }. Here every probem instance ceary is sovabe with optima vaue 1 (if a 11 1, then the optima soution is (1, 0), otherwise it is (0, 1)), whie the robust counterpart, which ceary is min x 1 + x 2 s.t. 1 2 x 1 + x 2 1 x 1 + 1 2 x 2 1 x 1 + x 2 = 1 x 1, x 2 0 is infeasibe. We are about to demonstrate that there are quite natura assumptions under which the indicated gap disappears, and the answers to the above two questions (A) and (B) become positive, so that (P U ) under these assumptions is not worse than the worst instance from P. The main part of the aforementioned assumptions is that the uncertainty shoud be constraintwise. To expain this concept, et U i be the set of a possibe reaizations of i-th row in the constraint matrix, i.e., be the projection of U R m n = R n... R n onto i-th direct factor of the right hand side of the atter reation 3). We say that the uncertainty is constraint-wise, if the uncertainty set U is the direct product of the partia uncertainty sets U i : By construction, x is r-feasibe if and ony if U = U 1 U 2... U m. a T x 0 a U i i; f T x = 1. (8) In other words, (P U ) remains unchanged when we extend the initia uncertainty set U to the direct product Û = U 1... U m of its projections on the sub-spaces of data of different constraints; the robust counterpart fees ony the possibe reaizations of i-th constraint, i = 1,..., m and does not fee the dependencies (if any) between these constraints in the instances. Thus, given an arbitrary uncertainty set U, we can aways extend it to a constraint-wise uncertainty set resuting in the same robust counterpart. In view of the above remarks, when speaking about questions (A) and (B) it is quite natura to assume constraint-wise uncertainty. An additiona technica assumption needed to answer affirmativey to (A) and (B) is the foowing: Boundedness Assumption.There exists a convex compact set Q R n which for sure contains feasibe sets of a probem instances (P ) P. 3) in order not to introduce both coumn and row vectors, we represent a row in an m n matrix by a coumn vector. 4
To ensure this assumption, it suffices to assume that the constraints of a instances have a common certain part (e.g., box constraints on the design variabes) which defines a bounded set in R n. Now we can formuate the main resuts of the subsection. Proposition 2.1 If the uncertainty U is constraint-wise and the Boundedness Assumption hods, then (i) (P U ) is infeasibe if and ony if there exists an infeasibe instance (P ) P. (ii) If (P U ) is feasibe and c is the optima vaue of the probem, then c = sup{c (P ) (P ) P}. (9) Proof. (i): since the feasibe set of (P U ) is contained in the feasibe set of every probem instance, the if part of the statement is evident. To prove the ony if part, assume that (P U ) is infeasibe, and et us prove that there exists an infeasibe instance (P ) P. Consider the system of inear inequaities (8) with unknown x and additiona restriction x Q, Q being given by the Boundedness assumption. Since (P U ) is infeasibe, this system has no soution. By the standard compactness arguments it foows that aready certain finite subsystem a T p x 0, p = 1,..., N, f T x = 1 of (8) has no soution in Q. Now et A 1,..., A N U be the instances from which the a p s come. We caim that the system of inequaities A 1 x 0,..., A N x 0, f T x = 1 (10) has no soutions. Indeed, by its origin it has no soutions in Q, and due to the Boundedness assumption it has no soutions outside Q as we. By Farkas Lemma, inconsistency of (10) impies existence of nonnegative λ ip, i = 1,..., m, p = 1,..., N, and a rea µ such that m N λ ip a p i p=1 + µf = 0, µ > 0, (11) where a p i is i-th row of A p. Let λ i = N λ ip. For a i with nonzero λ i, et a i be defined as a i = p=1 λ 1 N i λ ip a p i ; for i with zero λ i et a i = a 1 i. p=1 By construction, a i is a convex combination of i-th rows of certain instances, so that a i U i. Since the uncertainty is constraint-wise, the matrix A with the rows a T i, i = 1,..., m, beongs to U. On the other hand, (11) impies that m λ i a i + µf = 0 & µ > 0; but this means exacty that the probem instance given by A is infeasibe. Caim (ii) is an immediate consequence of (i). Indeed, et us denote the right hand side of (9) by d. Evidenty d c, and a we need to prove is that the strict inequaity here is impossibe. Assume, to the contrary, that d < c, and et us add to a our instances the common certain inequaity df T x c T x 0 (i.e., et us add an upper bound d on the objective vaue). The new uncertain LP probem ceary has constraint-wise uncertainty and satisfies the boundedness assumption, and due to the origin of d a its instances are feasibe. By (i), the robust counterpart (P U ) of this probem aso is feasibe. But this counterpart is nothing but (P U ) with added inequaity c T x d, and since d < c, (P U ) is infeasibe, which is the desired contradiction. 2.3 Computationa tractabiity of (P U ) Probem (P U ) can be equivaenty rewritten as min{c T x x G U }, G U = {x Ax 0 A U; f T x = 1}. (12) 5
It is ceary seen that G U is a cosed convex set, so that (P U ) is a convex program. It is known [8] that in order to minimize in a theoreticay efficient manner a inear objective over a cosed convex set G R n it suffices to equip the set G with an efficient separation orace. The atter is a routine which, given as input a point x R n, reports whether x G, and if it is not the case, returns a separator of x and G, i.e., a vector e x R n such that e T x > sup x G e T x. Thus, the question what are the geometries of the uncertainty set U which ead to a computationay tractabe robust counterpart (P U ) of the origina LP program (P ) can be reformuated as foows: when can G U be equipped with an efficient separation orace? Reca that when answering this question, without oss of generaity we may restrict ourseves to the case of cosed convex uncertainty sets U. An intermediate answer to our question is as foows: in order to equip G U with an efficient separation orace, it suffices to buid an efficient incusion orace an efficient routine which, given on input an x R n, verifies whether the convex set U(x) = {Ax A U} is contained in the nonnegative orthant R m +, and if it is not the case, returns a matrix A x U such that the point A x x does not beong to R m +. Indeed, given an incusion orace R, we can imitate a separation orace for G U as foows. In order to verify whether x G U, we first check whether f T x = 1; if it is not the case, then x ceary is not in G U, and we can take as a separator of x and G U either f or f, depending on the sign of the difference f T x 1. Now assume that f T x = 1. Let us ca R to check whether U(x) R m +. If it is the case, then x G U ; otherwise R returns a matrix A x U such that y = A x x R m +, so that at east one of the coordinates of the vector y, say, i-th, is negative. Now consider the inear form e T x x e T i A xx of x R n, where e i is the i-th standard orth of R m. By construction, at the point x = x this form is positive, whie at the set G U it ceary is nonpositive (indeed, if x G U, then Ax R m + for a A U and, in particuar, for A = A x ; consequenty, the vector y = A x x is nonnegative and therefore e T x x = e T j y 0). Thus, e x separates x from G U, as required. We see that in order to equip G U with an efficient separation orace, it suffices to have in our disposa an efficient incusion orace R. When does the atter orace exist? An immediate exampe is the one when U is given as a convex hu of a finite set of scenarios A 1,..., A M. Indeed, in this case to get R, it suffices, given an input x to R, to verify whether a the vectors A i x, i = 1,..., M, are nonnegative. If it is the case, then U(x) = Conv{A 1 x,..., A M x} is contained in R m +, and if for some i the vector A i x is not nonnegative, we can take A i as A x. The case in question is of no specific interest, since here (P U ) simpy is an LP program min{c T x A 1 x 0,..., A M x 0, f T x = 1}. In fact, basicay a we need to get an efficient incusion orace is computationa tractabiity of U U itsef shoud admit an efficient separation orace. Namey, we can formuate the foowing Tractabiity Principe. An efficient Separation orace for U impies an efficient Incusion orace and, consequenty, impies an efficient Separation orace for G U (and thus impies computationa tractabiity of (P U )). Note that the formuated statement is a principe, not a theorem; to make it a theorem, we shoud add a number of unrestrictive technica assumptions (for detais, see [8]). The principe is amost evident: in order to detect whether U(x) R m + for a given x, it suffices to sove m auxiiary convex programs min{e T i Ax A U}, i = 1,..., m, A being the design vector. (Indeed, if the optima vaues in a these programs are nonnegative, U(x) R m + ; if the optima vaue in i-th of the probems is negative, then any feasibe soution A i to the i-th probem with negative vaue of the objective can be taken as separator A x ). Now, our auxiiary probems are probems of minimizing a inear objective over a cosed convex set U, and we aready have mentioned that basicay a we need in order to sove efficienty a probem of minimizing a inear objective over a convex set is an efficient Separation orace for the set). 6
According to the Tractabiity principe, a reasonabe cosed convex uncertainty sets U ead to computationay tractabe (P U ), e.g., a sets given by finitey many efficienty computabe convex inequaities 4). Note, however, that computationa tractabiity is a theoretica property which is not exacty the same as efficient sovabiity in practice, especiay given the huge sizes of some rea word LP programs. In order to end up with practicay sovabe robust counterpart (P U ), it is highy desirabe to ensure a simpe anaytica structure of the atter probem, which in turn requires U to be reativey simpe. On the other hand, when restricting ourseves with too simpe geometries of U, we oose in fexibiity of the above approach in its abiity to mode diverse actua uncertainties. In our opinion, a reasonabe soution to the conficting goas is the one where U is restricted to be an eipsoida uncertainty, i.e., an intersection of finitey many eipsoids sets given by convex quadratic inequaities. Some arguments in favour of this choice are as foows: eipsoida uncertainty sets form reativey wide famiy incuding, as we sha see, poytopes (bounded sets given by finitey many inear inequaities) and can be used to approximate we many cases of compicated convex sets. an eipsoid is given parametricay by data of moderate size, hence it is convenient to represent eipsoida uncertainty as input; in some important cases there are statistica reasons which give rise to eipsoida uncertainty (see Section 4); ast (and most important), probem (P U ) associated with an eipsoida U possesses a very nice anaytica structure as we sha see in a whie, it turns out to be a conic quadratic program (CQP), i.e., a program with inear objective and the constraints of the type a T i x + α i B i x + b i, i = 1,..., M, (13) where α i are fixed reas, a i and b i are fixed vectors, and B i are fixed matrices of proper dimensions; stands for the usua Eucidean norm. Recent progress in interior point methods (see, e.g., [4]) makes it possibe to sove truy arge-scae CQP s, so that eipsoida uncertainty eads to practicay sovabe robust counterparts (P U ). 3 Probem (P U ) in the case of eipsoida uncertainty 3.1 Eipsoids and eipsoida uncertainties In geometry, a K-dimensiona eipsoid in R K can be defined as an image of K-dimensiona Eucidean ba under a one-to-one affine mapping from R K to R K. For our purposes this definition is not that convenient. On one hand, we woud ike to consider fat eipsoids in the space E = R m n of data matrices A usua eipsoids in proper affine subspaces of E (such an eipsoid corresponds to the case when we dea with partia uncertainty, e.g., some of the entries in A are certain ). On the other hand, we want to incorporate aso eipsoida cyinders sets of the type sum of a fat eipsoid and a inear subspace. The atter sets occur when we impose on A severa eipsoida restrictions, each of them deaing with part of the entries; e.g., an interva m n matrix A (U is given then by upper and ower bounds on the entries of the matrix). In order to cover a these cases, we define an eipsoid in R K as a set of the form U = {Π(u) Qu 1}, (14) 4) indeed, in order to get an efficient Separation orace for a set U defined by finitey many convex constraints g i(a) 0 with efficienty computabe g i ( ), it suffices, given A, to verify whether g i (A) 0 for a i; if it is the case, A U, otherwise the (taken at A) subgradient of the vioated constraint is a separator of A and U 7
where u Π(u) is an affine embedding of certain R L into R k and Q is an M L matrix. This definition covers a previousy discussed cases: when L = M = K and Q is nonsinguar, U is the standard K-dimensiona eipsoid in R K ; fat eipsoids correspond to the case when L = M < K and Q is nonsinguar; and the case when Q is singuar corresponds to eipsoida cyinders. From now on we sha say that U R m n is an eipsoida uncertainty, if A. U is given as an intersection of finitey many eipsoids: with expicity given data Q and Π ( ); B. U is bounded; U = k U(Π, Q ) (15) =0 C. [ Sater condition ] there is at east one matrix A U which beongs to the reative interior of every eipsoid U(Π, Q ), = 1,..., k: k u : A = Π (u ) & Q u < 1. 3.2 Probem (P U ) for the case of eipsoida uncertainty U Our current goa is to derive expicit representation of probem (P U ) associated with the uncertainty set (15); as we sha see, in this case (P U ) is a CQP. 3.2.1 Simpest cases Let us start with the simpest cases where U is a usua eipsoid or is a constraint-wise uncertainty with every constraint uncertainty set U i being an eipsoid. U is a usua eipsoid: U = {A = P 0 + k u j P j u T u 1}, where P j, j = 0,..., k, are m n j=1 matrices. Let us denote by r (j) i i-th row of P j (reca that we aways represent rows of a matrix by coumn vectors), and et R i be the n k matrix with the coumns r (1) i,..., r (k) i so that i-th row of Π(u) is exacty r (0) i + R i u. A point x R n is r-feasibe if and ony if f T x = 1 and, for a i = 1,..., m, the inner product of i-th row in Π(u) and x is nonnegative whenever u 1, i.e., [r (0) i ] T x + (R i u) T x 0 (u, u 1) [ i = 1,..., m]. In other words, x is r-feasibe if and ony if f T x = 1 and [ ] ] T x + u T Ri T x min u: u 1 [r (0) i We see that (P U ) is nothing but the CQP = [r (0) i ] T x R T i x 0, i = 1,..., m. min{c T x [r (0) i ] T x R T i x, i = 1,..., m; f T x = 1} (16) U = m U i is constraint-wise uncertainty with eipsoids U i : U i = {A A T e i = r i + R i u i for some u i R L i with u i 1, i = 1,..., m}; here e i are the standard orths in R m, r i R n and R i are n L i matrices. Exacty as above, we immediatey concude that in the case in question (P U ) is the CQP min{c T x r T i x R T i x, i = 1,..., m; f T x = 1}. (17) 8
The case of genera eipsoida uncertainty. In the case of genera eipsoida uncertainty, the robust counterpart aso turns out to be a conic quadratic program, i.e., a program with finitey many constraints of the form (13): Theorem 3.1 The robust counterpart (P U ) of an uncertain LP probem with genera eipsoida uncertainty can be converted to a conic quadratic program. Proof: see Appendix. Note that, as it was aready mentioned, conic quadratic probems can be soved by poynomia time interior point methods at basicay the same computationa compexity as LP probems of simiar size. Remark 3.1 It was mentioned earier that the case when the uncertainty set U is a poytope, i.e., U = {u R k d T i u r i, i = 1,..., M} [d i 0] (18) is in fact a case of eipsoida uncertainty. This is not absoutey evident in advance: a poytope by definition is an intersection of finitey many haf-spaces, and a haf-space is not an eipsoid an eipsoid shoud have a symmetry center. Nevertheess, a poytope is an intersection of finitey many eipsoida cyinders. Indeed, since U given by (18) is bounded, we can represent this set as an intersection of stripes : U = {u s i d T i u r i, i = 1,..., M}, ensuring the differences r i s i to be arge enough. And a stripe is a very simpe eipsoida cyinder : {u R k s i d T i u r i } = U(p i, I, Q i ), where p i is an arbitrary point with d T i p i = (s i + r i )/2, I is the unit k k matrix and Q i is the 1 k matrix (i.e., a row vector) given by Q i u = r i s i 2 d T i u. Thus, a poytopic uncertainty is indeed eipsoida. 4 Exampe It is time now to discuss the foowing important issue: where coud an eipsoida uncertainty come from? What coud be the ways to define the eipsoids constituting the uncertainty set? One natura source of eipsoida uncertainty was aready mentioned: approximation of more compicated uncertainty sets, which by themseves woud ead to difficut robust counterparts of the uncertain probem. We can hardy say anything more on this issue here everything depends on the particuar situation we meet. In case we are given severa primary scenarios of the data, we coud construct the uncertainty eipsoid as the minima voume eipsoid containing these scenarios (or a sma neighbourhood of this finite set). There is, however, another source of eipsoida uncertainties which comes from statistica considerations. To expain it, et us ook at the foowing exampe: A simpe portfoio probem. $1 is to be invested at the beginning of the year in a portfoio comprised of n shares. The end-of-the-year return per $1 invested in share i is p i > 0. At the end of the year you se the portfoio. The goa is to determine the amount x i to be invested in share i, i = 1,..., n, so as to maximize the end-of-the-year portfoio vaue n p i x i. When the quantities p i are known in advance, the situation is modeed by the foowing simpe LP program: n n max{ p i x i x i = 1, x i 0}, (19) 9
and the optima soution is evident: we shoud invest a we have in the most promising (with the argest p i ) share. Assuming that the coefficients p i are distinct and arranged in ascending order: p 1 < p 2 <... < p n, the soution is x n = 1, x i = 0, i = 1,..., n 1. Now, what happens if the coefficients p i are uncertain, as it is in reaity? Assume that what we know are nomina vaues of these coefficients p i, p 1 < p 2 <... < p n and bounds σ i < p i such that the actua vaues of p i are within the uncertainty intervas i = [p i σ i, p i + σ i]. Assume, moreover, that p i are of statistica nature and that they are mutuay independent and distributed in i symmetricay with respect to the nomina vaues p i. Under the indicated assumptions the simpest Stochastic Programming reformuation of the probem the one where we are interested to maximize the expectation of the fina portfoio vaue is given by the same LP program (19) with p i repaced by their nomina vaues p i ; the nomina soution is simpy to invest a the money in the most promising share n (x n = 1, x j = 0, j = 1,..., n 1). This poicy wi resut in random yied x nom with the expected vaue E{x nom } = p n. (20) Now et us ook what coud be done with the Robust Counterpart approach. The ony question here is how to specify the uncertainty set, and the most straightforward answer is as foows: in the situation in question, the vector p of uncertain coefficients runs through the box B = {(p 1,..., p n ) p i p i σ i i}, every point of the box being a possibe vaue of the vector. Thus, the uncertainty set is B (here this is exacty the approach proposed by Soyster). The corresponding robust optima poicy woud be to invest everything in the shares with the argest worst-case return p i σ i. Of course, this poicy is too conservative to be of actua interest. Note, however, that when appying the Robust Counterpart approach, we are not obiged to incude into the uncertainty set a which may happen. For the LP probem (19) and the underying assumption on the uncertain coefficients p i, we propose the uncertainty eipsoida set: U θ = {p R n n σ 2 i (p i p i ) 2 θ 2 }. The parameter θ is a subjective vaue chosen by the decision maker to refect his attitude towards risk; the arger is θ, the more risk averse he is. Note that for θ = 0, U θ shrinks to the singeton U 0 = {p } the nomina data; for θ = 1, U 1 is the argest voume eipsoid contained in the box B = {p p i p i σ i, i = 1,..., n}, and for θ = n, U θ is the smaest voume eipsoid containing the box. Writing the LP probem (19) in the equivaent form { } n n max y y p i x i, x 1 = 1, x 0, (21) we can use the resut of Section 3.2.1 to derive the foowing robust counterpart of (21) with respect to the uncertainty set U θ : { n } max p i x i θv 1/2 n (x) x i = 1, x 0, V (x) = n (22) σi 2x2 i. 10
Probem (22) resembes much the Markovitz approach to portfoio seection, athough in this cassica approach the roe of V 1/2 (x) is payed by V (x). The robust counterpart (22) can be motivated by reasoning as foows. Assume that we distribute the unit amount of money between the shares as x = (x 1,..., x n ). The corresponding yied y = n p i x i can be expressed as where the random part ζ has zero mean and variance n n y = p i x i + ζ, ζ = x i [p i p i ]. (23) n n Var(ζ) = (x i ) 2 E{(p i p i ) 2 } V (x) = (x i ) 2 σi 2. Consequenty, the typica vaue of y wi differ from the nomina vaue m p i x i by a quantity of order of Var 1/2 (ζ) V 1/2 (x), variations in both sides being equay probabe. A natura idea to hande the uncertainty is as foows. Let us choose somehow a reiabiity eve θ and ignore the events where the noise ζ is ess than θv 1/2 (x) 5). Among the remaining events we take the worst one ζ = θv 1/2 (x), and act as if it was the ony possibe event. With this approach, the stabe yied of a decision x is n p i x i θv 1/2 (x). These considerations ead precisey to the robust counterpart (22) obtained above. It shoud be stressed that the uncertainty eipsoid U θ is in no sense an approximation of the support of the distribution of p. Assume, e.g., that p i takes vaues p i ± σ i with probabiity 1/2 and that θ = 6 (the probabiity to get p i x i < p i x i θv 1/2 (x) is < 10 7 for every x 5) ). Under these i i assumptions, for not too sma n namey, for n > θ 2 = 36 the eipsoid U 6 does not contain a singe reaization of the random vector p! Note aso that the resuting uncertainty eipsoid depends on the safety parameter θ. In appications, a decision maker coud sove the robust counterparts of the probem for severa vaues of θ and then choose the one which, in his opinion, resuts in the best tradeoff between safety and greed. The above considerations iustrate how one can specify eipsoida uncertainty sets, starting from stochastic data. Now et us ook at probem (22). In order to demonstrate what coud be the effect of passing from the nomina program (19) with p i set to their nomina vaues p i to the probem (22), consider the foowing numerica exampe with n = 150. The nomina coefficients p i, i = 1,..., n, form an arithmetic progression: p i = λ + iδ, λ = 1.15, δ = 0.05 [p 1 1.15, p 150 = 1.2] 150 and the parameters σ i are chosen to be σ i = 1 3 δ 2in(n + 1) 0.0236 i [σ 1 = 0.0236, σ 150 = 0.2896]. Note that σ i are increasing, so that more promising investments are aso more risky. With the above data, we consider three candidate investment poicies: the nomina one, N, which is the optima soution of the nomina program (20), and cas for investing a we have in the most promising share. For this poicy the expected yied is y nom = 1.2, and the standard deviation of the yied is 0.290. Aso, with probabiity 0.5 we oose 5) It can be easiy seen that Pr{ζ < θv 1/2 (x)} < exp{ θ 2 /2}; for θ = 6 the atter quantity is < 10 7. 11
9% of the initia capita, and with the same probabiity this capita is increased by 45%; one hardy coud consider this unstabe poicy as appropriate 6) the robust counterpart one, RC, which is the optima soution of probem (22) with θ = 1.5. Here, due to our particuar choice of σ i, this poicy is to invest equay in a shares, i.e., x i = 1/n, i = 1,..., n = 150, the robust optima vaue being 1.15 7) ; the robust poicy of Muvey, Vanderbei and Zenios [11]; this poicy, MVZ, comes from the robust optima soution given by methodoogy of [11], i.e., the optima soution to the probem { max y µ N g(y x T p t ) N t=1 } n x i = 1, x i 0, (24) where p 1,..., p N are given scenarios reaizations of the yied coefficients p = (p 1,..., p n ), and the function g is a penaty function for vioating the uncertain constraint y x T p aong the set of scenarios. The parameter µ > 0 refects the decision maker tradeoff between optimaity and feasibiity. A typica choice of g in [11], which is aso used here, is g(z) = z + max[z, 0]. In our experiments µ = 100 and the scenario # t, p t = (p t 1,..., pt n), t = 1,..., N, is chosen at random from the distribution corresponding to i.i.d. entries p t i taking with probabiity 1/2 the vaues p i ± σ i. In order to compare the stabiity properties of the second and the third poicies (the characteristics of N poicy are cear in advance, see above), we set the number of scenarios N in MVZ to 1, 2, 4, 8,..., 256, then generated randomy the corresponding set of scenarios, and soved the resuting probem (24). Given the soution to the probem, we test it against the RC-soution, running 400 simuations of the random yieds p = (p 1,..., p n ) and comparing the random yieds given by the poicies in question. The resuts of the comparison are given in Tabe 1: N R ov M ov R mn M mn R mx M mx R av M av R sd M sd R s M s 1 1.15 1.49 1.13 0.91 1.23 1.49 1.18 1.20 0.017 0.277 0.00 50.25 2 1.15 1.48 1.12 0.92 1.22 1.48 1.18 1.20 0.017 0.277 0.00 49.75 4 1.15 1.45 1.11 0.93 1.22 1.45 1.18 1.19 0.017 0.263 0.00 49.00 8 1.15 1.45 1.13 0.93 1.23 1.45 1.18 1.19 0.017 0.263 0.00 48.25 16 1.15 1.32 1.12 0.93 1.23 1.45 1.18 1.19 0.017 0.128 0.00 7.00 32 1.15 1.24 1.13 0.98 1.23 1.38 1.18 1.19 0.017 0.069 0.00 0.50 64 1.15 1.20 1.13 1.03 1.22 1.32 1.18 1.18 0.017 0.044 0.00 0.00 128 1.15 1.18 1.13 1.08 1.23 1.28 1.18 1.18 0.017 0.037 0.00 0.00 256 1.15 1.17 1.12 1.08 1.23 1.26 1.18 1.18 0.017 0.030 0.00 0.00 Tabe 1: resuts for RC and MVZ soutions The headers in the tabe mean the foowing: N is the # of scenarios used in the MVZ-soution; prefixes R,M correspond to the RC and MVZ poicies, respectivey; the meaning of the symbos ov,mn,mx,av,sd is: ov - optima vaue mn - minima observed yied mx - maxima observed yied av - expected yied sd - standard deviation of yied s - empirica probabiity (in %) of yied < 1 6) Of course, there is nothing new in the phenomenon in question; OR financia modes take care not ony of the expected yieds, but aso of the variances and other characteristics of risk since the semina work of Markovitz [10]. Note that in the particuar exampe we treat here the robust counterpart soution resembes a ot the one given by the Markovitz approach. 7) For different data, the RC poicy gives different portions to the various shares, but sti impements the principe not to put a eggs in the same basket 12
We see from the resuts in Tabe 1 that, as far as the standard deviation is concerned, the RCpoicy is about 15 times more stabe than the nomina one; besides this, in a our 400 9 = 3, 600 simuations the RC-poicy never resuted in osses (an in fact aways yieded at east 11% profit), whie the nomina poicy resuts in 9% osses with probabiity 0.5. As about the MVZ-poicy, it is indeed more stabe than the nomina one, athough ess stabe than the RC-poicy. The stabiity properties of MVZ heaviy depend on the number of scenarios: with 16 scenarios, the poicy in 7% of cases resuts in osses and has standard deviation of the yied 7.5 times arger than for the RC-poicy. As the number of scenarios grow, the stabiity properties of MVZ approach those of RC, athough the standard deviation for MVZ remains 1.8 times worse than the one for RC even for 256 scenarios. Note aso that for sma numbers of scenarios MVZ is ess conservative than RC resuts in a better average yied; this phenomenon, however, disappears aready for 64 scenarios, and with this (and arger) # of scenarios MVZ is simpy dominated by RC both poicies resut in the same average yied, but the first one has worse standard deviation. We next study the behaviour of the optima soution to the RC probem (22) (denoted by x (θ)) as a function of the uncertainty parameter θ. Reca that the eipsoida uncertainty is the arger the arger is θ. In particuar, x (0) is the nomina soution. Three characteristics of the optima soution are particuary important: the expected profit π(θ) = n n p i x i (θ) 1, its standard deviation ν(θ) = σi 2(x i (θ))2 and the net robust optima vaue ρ(θ) the optima vaue in (22) minus 1. As we can see from Fig. 2, both π(θ) and ρ(θ), as we as ν(θ), vary sowy with θ, given θ 1. This impies that our nomina choice θ = 1.5 is not crucia to the type of resuts obtained for RC-poicy and reported in Tabe 1. 0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Figure 2. Characteristics of the RC poicy as functions of θ dotted: ρ(θ); soid: π(θ); dashed: ν(θ) The very simpe singe-stage mode with independent returns discussed here can be extended derive robust investment poicies for muti-stage Portfoio probem with dependent returns, see [5]. References [1] Ben-Ta, A. The Entropic Penaty approach to Stochastic Programming, Mathematics of Operations Research, v. 10 (1985), 263-279. [2] Ben-Ta, A., and Nemirovski, A. Robust Truss Topoogy Design via Semidefinite Programming SIAM Journa of Optimization v. 7 (1997), 991-1016 13
[3] Ben-Ta, A., and Nemirovski, A., Robust Convex Optimization Mathematics of Operations Research, to appear in Nov. 1998. [4] Ben-Ta, A., and Zibuevsky, M. Penaty/barrier mutipier methods for convex programming probems, SIAM Journa of Optimization v. 7 (1997), 347-366. [5] Ben-Ta, A., Margeit, T., and Nemirovski, A., Robust Modeing of Muti-Stage Portfoio Probems, to appear in Proceedings of Workshop on High-Performance Optimization, Rotterdam, August 1997, Kuwer Academic Press. [6] Birge, J.R., and Louveaux, F. Introduction to Stochastic Programming, Springer, 1997. [7] Fak, J.E. Exact Soutions to Inexact Linear Programs Operations Research (1976), pp. 783-787. [8] Grötsche, M., Lovasz, L., and Schrijver, A. The Eipsoid Method and Combinatoria Optimization, Springer, Heideberg, 1988. [9] Ka, P., and Waace, S.W. Stochastic Programming, Wiey-Interscience, New York, 1994 [10] Markovitz, H.M., Portfoio Seection: Efficiency Diversification of Investment Wiey, New York, 1959 [11] Muvey, J.M., Vanderbei, R.J. and Zenios, S.A. Robust optimization of arge-scae systems, Operations Research 43 (1995), 264-281. [12] Nesterov, Yu., and Nemirovski, A. Interior Point Poynomia Agorithms in Convex Programming SIAM Studies in Appied Mathematics, SIAM, Phiadephia, 1994. [13] Prékopa, A. Stochastic Programming, Kuwer Academic Pubishers, 1995. [14] Rockafear, R.T., and Wets, R.J.-B., Scenarios and poicy aggregation in Optimization under uncertainty, Mathematics of Operations Research v. 16 (1991), 119-147. [15] Singh, C. Convex Programming with Set-Incusive Constraints and its Appications to Generaized Linear and Fractiona Programming Journa of Optimization Theory and Appications, v. 38 (1982), No. 1, pp. 33-42. [16] Soyster, A.L. Convex Programming with Set-Incusive Constraints and Appications to Inexact Linear Programming - Operations Research (1973), pp. 1154-1157. [17] Zhou, K., Doye, J.C., and Gover, K. Robust and Optima Contro, Prentice Ha, 1996. Appendix: Proof of Theorem 3.1 Let U be a genera eipsoida uncertainty (15): U = k U(Π, Q ), U(Π, Q ) = {A = Π (u ) Q u 1}. [A R m n ] =0 Let a T i [A] be i-th row of a matrix A. We first observe that 14
(I) A point x R n satisfying the normaization constraint f T x = 1 is robust feasibe if and ony if for every i = 1,..., m the optima vaue in the optimization probem a i [Π 0 (u 0 )] T x min s.t. { Π (u ) = Π 0 (u 0 ), = 1,..., k, Q (u ) 1, = 0, 1,..., k (P i [x]) with design variabes u 0,..., u k is nonnegative. Indeed, it is readiy seen that U is exacty the set of vaues attained by the matrix-vaued function Π 0 (u 0 ) at the feasibe set of (P i [x]). Now, (P i [x]) is a conic quadratic optimization probem one of the generic form { e T z + φ min s.t. Rz = r, A z b c T z d, = 0,..., k (CQP p ) (the design vector in the probem is z, whie A, b, c, d, = 0,..., k, e, φ are given matrices/vectors/scaars). It is known ([12], Chapter 4) that the Fenche dua of (CQP p ) is again a conic quadratic probem, namey, the probem r T λ + k [d ν + b T µ ] + φ max s.t. =0 R T λ + k [ν c + A T µ (CQP ] = e, d ) =1 µ ν, = 0,..., k. with design variabes λ, {ν, µ } k =0 which are scaars (ν ) or vectors (λ, µ ) of proper dimensions. Moreover, the Fenche duaity theorem in the case in question becomes the foowing statement (see [12], Theorem 4.2.1): (II) Let the prima probem (CQP p ) be stricty feasibe (i.e., admitting a feasibe soution ẑ with A ẑ b < c T ẑ d, = 0,..., k) and et the objective of (CQP p ) be bounded beow on the feasibe set of the probem. Then the dua probem (CQP d ) is sovabe, and the optima vaues in the prima and the dua probems are equa to each other. Note that when (CQP p ) is the probem (P i [x]) (so that ony the objective in the probem depends on the parameter x, and this dependence is affine), then (CQP d ) becomes the probem of the form where λ (i), {µ (i), ν(i) } k =0 [r (i) ] T λ (i) + k [d (i) ν(i) + [b (i) =0 [R (i) ] T λ (i) + k [ν c (i) =0 µ (i) ]T µ (i) + [A (i) ] + φ(i) [x] max s.t. ]T µ (i) ν (i), = 0,..., k, are the design variabes of the probem, ] = e(i) [x], e (i) [x], φ (i) [x] are affine vector-vaued, respectivey, scaar functions of x, (D i [x]) r (i), R (i), {A (i), b(i), c(i), d(i) }k =0 are independent of x matrices/vectors/scaars readiy given by the coefficients of the affine mappings Π ( ), Q ( ). 15
Now, it is readiy seen that the assumptions B, C (see the definition of eipsoida uncertainty) ensure that the probem (P i [x]) is stricty feasibe and bounded beow, whence by (II) the optima vaue in (P i [x]) is equa to the one in (D i [x]). Combining this observation with (I), we see that (III) A vector x R n, f T x = 1, is robust feasibe, the uncertainty set being U, if and ony if for every i = 1,..., m there exist λ (i), {µ (i), ν(i) } k =0 satisfying, aong with x, the system of constraints [r (i) ] T λ (i) + k [d (i) ν(i) =0 [R (i) ] T λ (i) + k [ν c (i) =0 µ (i) We concude that (P U ) is equivaent to the probem + [b (i) ]T µ (i) ] + f (i) [x] 0, + [A (i) ]T µ (i) ν (i), = 0,..., k. ] = e(i) [x], (C i ) c T x min s.t. (x, λ (i), {µ (i), ν(i) } k =0 ) satisfy C i, i = 1,..., m, f T x = 1, (CQP) with design variabes x, {λ (i), {µ (i), ν(i) } k =0 }m. Specificay, x is robust feasibe if and ony if it can be extended to a feasibe soution of (CQP). To compete the proof of Theorem 3.1, it remains to note that (CQP) is a conic quadratic probem, and that the data specifying this probem are readiy given by the coefficients of the affine mappings Π ( ), Q ( ). Remark 4.1 The outined proof demonstrates that Theorem 3.1 admits severa usefu modifications, in particuar the foowing two: (i) Assume that the uncertainty set U is poyhedra: U = {A R m n u R k : A = Π(u), Q(u) 0}, where Π( ) : R m R m (n+1), Q( ) : R k R are affine mappings. Then (P U ) is equivaent to an LP program with sizes which are poynomia in m, n, k, and the data readiy given by the coefficients of the affine mappings Π( ), Q( ). (ii) Assume that the uncertainty set U is semidefinite-representabe: U = {A R m n u R k : A = Π(u), Q(u) 0}, where Π(u) is affine, Q(u) is an affine mapping taking vaues in the space of symmetric matrices of a given row SIZE L and B 0 means that B is a symmetric positive semidefinite matrix. Assume aso that U is bounded, and that there exists û with positive definite Q(û). Then (P U ) is equivaent to a semidefinite program, i.e., program of the form dim e T z z min s.t. A 0 + z i A i 0. (SDP) The sizes of (SDP) (i.e., dim z and the row sizes of A i ) are poynomia in m, n, k,, and the data e, A 0,..., A dim z of the probem are readiy given by the coefficients of the affine mappings Π( ), Q( ). 16