Robust Sensitivity Analysis for Linear Programming with Ellipsoidal Perturbation

Transcription

1 Robust Sensitivity Anaysis for Linear Programming with Eipsoida Perturbation Ruotian Gao and Wenxun Xing Department of Mathematica Sciences Tsinghua University, Beijing, China, September 27, 2017 Abstract Given an originay robust optima decision and aowing perturbation parameters of the inear programming probem to run through a imum uncertainty set controed by a variabe of perturbation radius, we do robust sensitivity anaysis for the robust inear programming probem in two scenarios. One is to keep the origina decision sti robust optima, the other is to ensure some properties of the origina decision preserved in the robust optima soution set. In each scenario, we do anayses in three cases with different perturbation styes. A modes in our study are formuated into either inear programs or second-order conic programs except for some cases considering more than one row perturbation in the constraint matrix. For those, we deveop a binary search agorithm. Keywords: Linear programming, sensitivity anaysis, robust optimization, second-order cone programming. 1 Introduction In this study, the perturbed inear programming probem is as foows: min s.t. c T 0 x Ax b, (A, b) U(), x 0, (1.1) 1

2 where x R n is a variabe, c 0 R n is fixed and (A, b) R m n R m is in an eipsoida uncertainty set U() determined by a variabe of perturbation radius (radii) that is nonnegative. Without oss of generaity, we do not consider uncertainty in the coefficients of the objective function since we can equivaenty reformuate the probem by introducing a new variabe t, repacing the objective with min t and adding a new constraint c T 0 x t. In this study, we mainy discuss three perturbation styes that the right-hand-side vector is perturbed hoisticay, the constraint matrix is perturbed row-wisey and hoisticay, respectivey. Here we wi not consider the simutaneous perturbation of the constraint matrix and the right-hand-side vector since we can reform it into min s.t. c T 0 x ( A ) b x 0, (A, b) U(), y (1.2) y = 1, x 0, where perturbation ony appears in the constraint matrix. The uncertainty set U() in those three styes is given as beow. Case 1: right-hand-side vector perturbed hoisticay U() = {A 0 } U b = {A 0 } b Rm b = b 0 + Case 2: constraint matrix perturbed row-wisey, where = ( 1, 2,..., m ) T, t β j v j, β. (1.3) j=1 U() = U A1 {b 0 } = A n i Rm n A i = A i 0 + ( αju i i j) T, α i i, i = 1,, m {b 0}. j=1 (1.4) Case 3: constraint matrix perturbed hoisticay U() = U A2 {b 0 } = A Rm n A = A 0 + s γ j A j, γ {b 0}. (1.5) j=1 2

3 Here β = (β 1,, β t ) T, α i = (α1 i,, αi n i ) T, i = 1,, m and γ = (γ 1,, γ s ) T are perturbation parameters taking vaues arbitrariy in some bas. A i 0 is the i-th row of A 0 and A i is the i-th row of A i. v j, j = 1,, t; u i j, j = 1,, n i, i = 1,, m are perturbation direction vectors and A j, j = 1,, s are perturbation direction matrices. A of them are given by experience and estimation with no distributiona assumption on themseves, since we do not have numerous data to obtain a probabiistic description of the uncertainty set, which is different from stochastic programming referred in [7], [8] and [11]. Notice that is a vector in the second case and is a number in the other two cases. In case of U() = {A 0 } {b 0 } when = 0, (1.1) is indeed the origina inear programming probem which is supposed to be sovabe. In cassica sensitivity anaysis for the inear programming probem, the ranges of the objective coefficients and the ranges of the right-hand-side vector are anayzed to keep the optima soution unchanged or the optima basis unchanged, respectivey, which can be easiy found in books about inear programming. In both cases, the ranges of the parameters are given in a "box" form. However, an optima soution to one inear program with the parameters fixed may be infeasibe for another reaization of the inear program in the cassica sensitivity anaysis. To get a soution capabe of coping with a possibe reaizations of the parameters, the robust optimization mode is a promising choice. When is fixed, (1.1) is the robust optimization counterpart of inear programming. Robust optimization which came from the robust contro community (refer to [9] and [14]) dates back to the work in [12], and has deveoped rapidy in the past four decades, see e.g., [6], [2] and the references therein. The form of uncertainty sets, the performance assessment of robust optima soutions and the computationa compexity of the modes are three main topics considered in the robust optimization modeing. Based on the basic concept of the robust optimization, the uncertainty set(s) shoud contain a the observed data. So the easiest one is an interva by providing the minimum and imum of the data for each individua parameter. The uncertainty sets ike eipsoids or poytopes are then used to describe those data. For the interva sets, [5] found the traditiona robust optimization mode is rather conservative for it is sensitive to the uncertain data and even has no feasibe soution sometimes. They gave a modified mode with a probabiistic guarantee of feasibiity for its robust optima soution by counting partia uncertain data of the interva form. With the same concerns on feasibiity, [4] estabished convex uncertainty sets ike poytopes and eipsoids by using distortion measures 3

4 to evauate the feasibiity. A modes in [4] and [5] preserve computationa tractabiity which are poynomiay sovabe. But any robust optima soution of the modes in [4] and [5] may not be feasibe with a probabiity for the origina robust probems, which somehow does not satisfy the concept of robust optimization. Now we consider the robust optimization in a different way. When the uncertainty set is described by the eipsoida form, the robust inear programming is poynomiay computabe for any given, see [1]. Then for any robust optima soution seected with a given, is this given soution sti robust optima for a arger data perturbation? Under the assumption of the uncertainty set of eipsoids, the radius of any eipsoid is regraded as a variabe and the imum radii shoud be cacuated to obtain the imum data perturbation. This is our robust sensitivity anaysis concept. More concretey, two scenarios are considered in this study. One is to keep the originay robust optima soution x sti robusty optima, the other is to keep 0 entries of x preserved in a robust optima soution set. These two scenarios reay make sense in decision-making of practica production process, investment and other fieds. For exampe, we have got the optima strategy for the current stage. In the coming stage, we just have od information and we have to make a decision before parameters are exacty known. Of course, it woud be the best if the robust optima soution is sti x since we need not change the decision at a, which corresponds to the first scenario. Or at east, it is an acceptabe decision that the items not considered in the current stage wi not be considered in the coming stage, avoiding new setup costs, which corresponds to the second scenario. For simpicity, we ca the two scenarios as "optima soution unchanged" and "0 entries unchanged" respectivey. In our study, two main modes are provided associated with the two scenarios. Mode 1 for the first scenario is: s.t. x OPT 1 (), 0, (1.6) 4

5 where OPT 1 () is the optima soution set of (1.1). Mode 2 for the second scenario is: s.t. OPT 2 (), (1.7) 0, where OPT 2 () is the optima soution set of the foowing perturbed probem derived from (1.1) with an additiona constraint x j = 0, j P, where P = {j x j determined by the given decision x, = 0} is a given index set min s.t. c T 0 x Ax b, (A, b) U(), x 0. (1.8) Mode 1 is to find the imum of radius (radii) to keep the originay optima soution x sti robust optima to (1.1). Mode 2 is to find the imum of radius (radii) to ensure the robust optima soution set of (1.8) nonempty. [15] presented copositive programs to state the best-case and worst-case optima vaues when the coefficients in the objective function and the right-hand side are perturbed. And they deveoped tight, tractabe conic reaxations to provide the ower and upper bounds. They caed their work as "robust sensitivity anaysis of the optima vaue of inear programming". However, their idea is quite different from ours. Their uncertainty set is fixed rather than variabe, their feasibe soution set is not suitabe for a possibe reaizations of the parameters and perturbation ony in objective and right-hand side is aowed, whie the constraint matrix is not considered. Our main contributions are stated as foows. First, to the best of our knowedge, this robust sensitivity anaysis concept is first brought up by us and can be appied to evauate the robustness of a pre-decision. Second, we estabish modes corresponding to two scenarios, which shows the possibiity of appying our concept. The modes are either inear or bi-convex programs for the three perturbation cases. The bi-convex programs can be equivaenty formuated into second-order conic programs when ony one row perturbation is considered. Whie 5

6 for those modes considering more than one row perturbation, we haven t found equivaent poynomia ones and thus provide a binary search agorithm, which triggers our interest in the bi-convex optimization probems. Now we provide some notations needed in this paper. R n denotes the set of n dimensiona rea vectors, R n + denotes the set of n dimensiona nonnegative rea vectors. 2-norm is used in this paper, namey, x 2 = ( n i=1 x2 i )1/2, where x R n, and causing no confusion, we omit the subscript 2 afterwards. And x T y = n i=1 x iy i, where x, y R n. In addition, a the vectors mentioned in this paper are coumn vectors. The rest of the paper is organized as foows. In Section 2 and Section 3, we do robust anaysis in two scenarios respectivey, and in each scenario, three cases are considered. In Section 4, numerica experiment is showed. In addition, two notes and the concusions are given in Section 5 and Section 6. 2 Optima Soution Unchanged This section gives anaysis of the first scenario (1.6) in the three cases. In our assumption, x is supposed to be a robust optima soution for a given. We wi get the imum radius (radii) of, no smaer than, such that x is sti robust optima. For simpicity, et = 0 in the foowing consideration. 2.1 b 0 Perturbed Hoisticay Case 1 is considered in this subsection and the perturbed probem is min c T 0 x t s.t. A 0 x b 0 + β j v j, β, j=1 (2.1) x 0. where β = (β 1,, β t ) T R t runs through the ba centered at the origin with a radius. Let v i R t consists of a the i-th entries of v j, j = 1,, t. Then 6

7 x is feasibe for (2.1) (A 0 x b 0 ) i min β βt v i, i = 1,, m, and x 0 (A 0 x b 0 ) i v i, i = 1,, m, and x 0. Denote v = ( v 1,, v m ) T R m +. (2.1) is equivaent to min s.t. c T 0 x A 0 x b 0 v, x 0. (2.2) Now in Mode 1, OPT 1 () stands for the optima soution set of (2.2). Lemma 1. For x, feasibiity is enough to promise optimaity. Proof: Since and v are both nonnegative, we have A 0 x b 0 for any x feasibe for (2.2), thus x F 0. Then we have c T 0 x c T 0 x because x is originay optima. So once x is feasibe for (2.2), it is optima. Then (1.6) of Mode 1 is formuated as s.t. A 0 x b 0 v, 0. (2.3) Remark 1. The feasibe region of in (2.3) is an interva whose eft end point is zero. The foowing resuts are obvious by Lemma 1 and (2.3). Theorem 1. For (2.3), (i) if there exists i such that (A 0 x b 0 ) i = 0 and v i > 0, the imum radius = 0; (ii) if (i) does not hod for any i = 1,, m, denote J = {i v i > 0}. If J is empty, =, otherwise = min i J (b 0 A 0 x ) i / v i > 0. In case of (i) above, the constraint is caed active. Athough β cannot take vaues arbitrariy in a ba with a positive radius for an active constraint, it can vary in certain directions. For some 7

8 i which satisfies (A 0 x b 0 ) i = 0 and v i > 0, the inequaity β T v i 0 must hod, which means the ange between β and v i is no arger than π/2. Then D β = { β β T } v i 0, i I 1 stands for an aternative direction set of β, where I 1 = {i (A 0 x b 0 ) i = 0 and v i > 0}. So we can ony seect β in the set D β if it is nonempty. Denote I 2 = {i (A 0 x b 0 ) i < 0 and v i > 0}. If I 2 is nonempty, the imum norm 1 of β is min i I 2 (b 0 A 0 x ) i / v i when β is seected arbitrariy in D β, otherwise 1 = +. Now the region where β can vary arbitrariy is a part of a ba rather than a whoe ba. Obviousy, the probem cannot be perturbed at a when D β is empty. The resut that any inear programming probem can get its optimum at boundary points impies that (i) happens generay. In addition to treating case (i) ike above, we can introduce a toerance δ > 0 to (b 0 ) i and the constraint becomes (A 0 x b 0 ) i δ. Now the parameters in the i-th constraint can be perturbed and the imum perturbation radius is δ/ v i > 0. Of course, a bigger toerance we permit, a bigger perturbation radius we wi obtain. In practica production, a toerance is reasonabe since the manufacturer had better repenish one resource when it is used up exacty in the current stage. Here δ can be regarded as the extra suppement quantity of this resource. 2.2 A 0 Perturbed Row-wisey Case 2 is considered in this subsection and the perturbed probem is min c T 0 x n i s.t. (A i 0 + ( αju i i j) T )x b i 0, α i i, i = 1,, m, j=1 (2.4) x 0, where α i = (α i 1,, αi n i ) T R n i runs through the ba centered at the origin with radius i. Here A i 0 is the i-th row of A 0 and b i 0 is the i-th entry of b 0. It is equivaent to min c T 0 x s.t. A i 0x + i H i x b i 0, i = 1,, m, (2.5) x 0, 8

9 where H i = (u i 1,, ui n i ) T R n i n. Now OPT 1 () stands for the optima soution set of (2.5). Simiary, feasibiity promises optimaity. Then (1.6) of Mode 1 is formuated as s.t. A i 0x + i H i x b i 0, i = 1,, m, (2.6) 0. Here = ( 1,, m ) T, thus this is a muti-objective program. Since i, i = 1,, m are independent, imizing is equivaent to imizing a i, i = 1,, m. Then we can divide (2.6) into m subprobems and consider the foowing subprobem for each constraint i, i s.t. A i 0x + i H i x b i 0, (2.7) i 0. Theorem 2. For (2.7), if H i x = 0, the imum radius i = +, otherwise i = (b i 0 Ai 0 x )/ H i x. It is ucky to find that = ( 1,, m ) T is an absoute optima soution to the mutiobjective programming probem (2.6), namey, for any feasibe soution, we have. Simiary, we can introduce a toerance δ > 0 for those active constraints to avoid unperturbationa cases. 2.3 A 0 Perturbed Hoisticay Case 3 is considered in this subsection and the perturbed probem is min c T 0 x s s.t. (A 0 + γ j A j )x b 0, γ, j=1 (2.8) x 0, 9

10 where γ = (γ 1,, γ s ) T R s runs through the ba centered at the origin with radius. Denote U i = (A it 1,, AiT s ) T R s n, i = 1,, m. Then (2.8) is equivaent to min c T 0 x s.t. A i 0x + U i x b i 0, i = 1,, m, (2.9) x 0. Simiary, (1.6) of Mode 1 is formuated as s.t. A i 0x + U i x b i 0, i = 1,, m, (2.10) 0. Theorem 3. For (2.10), if U i x = 0, i = 1,, m, the imum radius = +, otherwise = min i K (b i 0 Ai 0 x )/ U i x, where K = {i U i x > 0}. Simiary, we can introduce a toerance δ > 0 for those active constraints to avoid unperturbationa cases. 3 0 Entries Unchanged This section gives anaysis of the second scenario (1.7) in the three cases. When is a vector, Mode 2 is a muti-objective programming. Generay, there does not necessariy exist an absoute optima soution to a muti-objective programming, refer to [10]. See the foowing exampe. Exampe 1. min 2x 1 x 2 + 2x 3 s.t. 3x 1 + 4x 2 + x 3 2, x 1 3x 2 2x 3 1, (3.1) x i 0, i = 1,, 3. 10

11 An optima soution x = (0.4, 0.2, 0) T of the exampe above is seected and P = {3} here. The perturbation direction vectors for the first row are u 1 1 = (0.1, 0.1, 0.2)T and u 1 2 = (0.3, 0.2, 0.1)T, and the one for the second row is u 2 1 = ( 0.1, 0.2, 0.1)T, then 0.1 H 1 = , H 2 = ( ) (3.2) Then (1.7) is formuated as foows: s.t. 3x 1 + 4x 2 + x H 1 x 2, x 1 3x 2 2x H 2 x 1, (3.3) x 3 = 0, 0, x i 0, i = 1, 2, 3, where = ( 1, 2 ) T. First, we ony perturb the first row, namey et 2 = 0. Actuay, (3.3) with 2 = 0 can be reformed into a second-order cone program which wi be proved ater, and the imum 1 is eading to a feasibe soution = (8.9443, 0) T of (3.3). Simiary, the imum 2 is when 1 is fixed to be 0 so = (0, ) T is aso feasibe for (3.3). Henceforth, if there exists an absoute optima soution, we must have (8.9443, 0) T and (0, ) T. However, even the most promising choice = (8.9443, ) T fais to be feasibe. Therefore, we wi provide two indirect modes beow to turn a muti-objective programming into a singe-objective programming and wi discuss them in detai in subsequent subsections. Preference mode: k s.t. OPT 2 (), = k g, (3.4) k 0, where k is a seected index, and g R m + is a given vector with g k = 1. g i, i k represents 11

12 the weight of the i-th row compared with the k-th row. For exampe, if we hope the i-th row has a greater space to be perturbed than the k-th row, we can et g i > 1. The method is used when we have a preference on the rows. Maxmin mode: min i=1,,m i s.t. OPT 2 (), (3.5) 0. The method is used when we expect to make the minimum radius as arge as possibe. 3.1 b 0 Perturbed Hoisticay With amost the same argument of getting (2.2), the perturbed probem is min s.t. c T 0 x A 0 x b 0 v, x 0. (3.6) Now in Mode 2, OPT 2 () stands for the optima soution set of (3.6). Since the probem is attainabe if feasibe, (1.7) of Mode 2 is formuated as the foowing inear program: s.t. A 0 x b 0 v, x 0, 0. (3.7) 12

13 3.2 A 0 Perturbed Row-wisey With amost the same argument of getting (2.5), the perturbed probem is min c T 0 x s.t. A i 0x + i H i x b i 0, i = 1,, m, (3.8) x 0, Then (1.7) of Mode 2 is formuated as s.t. A i 0x + i H i x b i 0, i = 1,, m, (3.9) x 0, 0, where = ( 1,, m ) T R m. (3.9) cannot be soved easiy owing to the cross terms i H i x, i = 1,, m. To anayze this probem, we need an assumption here. Assumption 1. The feasibe region of the probem (3.8) with = 0 is bounded and has a strict interior point. Here a strict interior point means there exists a feasibe soution denoted as x such that A i 0 x < bi 0, i = 1,, m. Obviousy, it is required that the prima unperturbed probem (1.1) with = 0 has an interior feasibe soution under this assumption. As stated before, (3.9) may not have an absoute optima soution, thus the first indirect mode (3.4) is formuated as k s.t. A i 0x + g i k H i x b i 0, i = 1,, m, (3.10) x 0, k 0. Here k {1,, m} and g R m + is given with g k = 1. Denote Supp = {i g i > 0}. 13

14 Proposition 1. The optima vaue of (3.10) is stricty arger than 0. Proof: By Assumption 1, there exists x such that A i 0 x < bi 0, i = 1,, m, x j = 0, j P and x 0, denote W = {i H i x > 0} {1,, m}. If Supp W, et = min { (b i 0 A i 0 x)/(g i H i x ), i Supp W } > 0, (3.11) then is feasibe for (3.10) and the optima vaue is arger than 0. If Supp W =, ( x, ) is feasibe for (3.10) for any > 0, then the optima vaue is infinity which is arger than 0. Therefore, the probem can reay be perturbed. In order to determine when the imum perturbation radius of (3.10) is finite, we have the foowing theorem. Theorem 4. The optima vaue of (3.10) is finite if and ony if the foowing probem has a positive optima vaue. min Hx s.t. A 0 x b 0, (3.12) x 0, where H R ( i Supp n i) n consists of H i, i Supp in turn. Proof: On one hand, if the optima vaue of (3.12) is 0, then there exists an x such that A i 0 x bi 0, i = 1,, m, x 0 and H i x = 0, i Supp. Then ( x, ) is feasibe for (3.10) for any > 0 eading to an infinite optima vaue of (3.10), which causes a contradiction. On the other hand, if the optima vaue of (3.10) is infinite, et { (p), p = 1, } be a positive series with (p) +. Then for any (p), there exists x (p) such that x (p) j = 0, j P, x (p) 0, A i 0 x(p) b i 0, i / Supp and H ix (p) (b i 0 Ai 0 x(p) )/g i (p), i Supp. By the boundedness of Assumption 1 which is equivaent to that the feasibe region of (3.10) with k = 0 is bounded, the feasibe set reated to x of (3.10) is bounded and cosed. Then there exists M > 0 such that b i 0 Ai 0x M, i = 1,, m hod for any feasibe soution x of (3.10). Now H i x (p) M/g i (p), i Supp. Since the feasibe region of (3.10) is bounded and 14

15 cosed, we can seect a convergent subsequence from { x (p), p = 1, }, which is aso denoted as { x (p), p = 1, } for convenience with H i x (p) 0, i Supp and x (p) x 0. It is obvious that x 0 is aso feasibe for (3.10) with H i x 0 = 0, i Supp, which eads to a contradiction to a positive optima vaue of (3.12). (3.10) is a bi-convex optimization probem as it is a second-order cone program when is fixed and is a inear program when x is fixed. Generay a bi-convex optimization probem is intractabe. Next we consider a simpe case g k = 1, g i = 0, i k, where the perturbation ony appears in the k-th row, which is actuay poynomiay computabe. Theorem 5. Under Assumption 1, et g k = 1, g i = 0, i k in (3.10), then the optima vaue of (3.10) is infinity if the optima vaue of (3.12) is 0 and is finite otherwise. In case of finite optima vaue, (3.10) and the foowing second-order cone program share the same optima vaue. p s.t. H k q 1, p = b k 0s A k 0q, A 0 q b 0 s, (3.13) q j = 0, j P, s 0, q 0, where q = (q 1,, q n ) T R n, p R, s R. Proof: The first caim is obvious by Theorem 4. In case of the finite optima vaue of (3.10), we have H k x > 0 for any feasibe soution x of (3.12), and then t H k x > 0 for any feasibe soution x of (3.10) as the feasibe set of (3.12) concudes that of (3.10). Thus the foowing 15

16 probem is we defined and is an equivaent reformuation of (3.10) with g k = 1, g i = 0, i k. y/t s.t. H k x t, y = b k 0 A k 0x, A 0 x b 0, (3.14) x 0, t 0. Then we are to prove (3.14) is equivaent to (3.13). On one hand, for any feasibe soution (x, y, t) of (3.14), t > 0 hods. Thus, (q, p, s) = (x/t, y/t, 1/t) is we defined and is feasibe for (3.13). Therefore, the optima vaue of (3.13) is no ess than that of (3.14). On the other hand, there exists a feasibe soution (x 0, y 0, t 0 ) of (3.14) by the strict interior point assumption such that y 0 /t 0 > 0. Then for any feasibe soution (q, p, s) of (3.13), if p = 0, y 0 /t 0 > p = 0 obviousy hods; if p > 0, we caim that s > 0 hods. Otherwise, we have H k q 1, p = A k 0 q, A 0q 0, q j = 0, j P and q 0. And for any d > 0, (x (d), y (d), t (d) ) = (dq + x 0, dp + y 0, d + t 0 ) is feasibe for (3.14). Based on Assumption 1, y is aso bounded due to the boundedness of x and the second constraint in (3.14). Since d can be taken arbitrariy arge, we must have q = 0 and p = 0, which causes a contradiction to p > 0. Since s > 0, (x, y, t) = (q/s, p/s, 1/s) is we defined and is feasibe for (3.14) with y/t = p. As a consequence, the optima vaue of (3.14) is no ess than that of (3.13). Henceforth, the two probems share the same optima vaue. Theorem 5 presents a poynomiay computabe specia case for Mode 2, in which perturbation is considered in one row aone. However, we haven t found a poynomia formuation for a genera g. Thus we provide a binary search agorithm to sove it. Obviousy, the feasibe region of k is an interva whose eft end point is 0. Notice that once we obtain a strict interior point x in Assumption 1, we can provide a positive ower bound for the optima vaue. Then the agorithm is designed as foows: Here ε is a given precision. The number of iterations is no more than og 2 (1/(bε)) and it costs poynomia time to sove a second-order cone program in each iteration. 16

17 Agorithm: Step1: Sove (3.12), if the optima vaue is 0, the imum radius is k = + and goto Step4, otherwise goto Step2. Step2: Obtain an interior point x 1 and a positive ower bound b for the imum radius given by (3.11). And Let p = 0 and q = 1/b. Step3: Let k = 2/(p + q) in (3.10). If (3.10) is feasibe, et q = 1/ k, otherwise et p = 1/ k. If q p < ε, goto Step4, otherwise continue. Step4: Output the imum radius k. 1 To obtain an interior point, we just need to sove (3.8) with = 0 and the objective function changed to 0. Since the buit-in agorithm in MATLAB for inear programming is interior point agorithm, it wi return the anaytica center of the feasibe region, which is certainy an interior point. For the min method (3.5), we get s.t. A i 0x + i H i x b i 0, i = 1,, m, (3.15) i, i = 1,, m, x 0, 0. By the foowing Remark 2, we get the same agorithm above to sove (3.15). Remark 2. When is fixed, (3.15) is feasibe if and ony if (3.15) is feasibe with i =, i = 1,, m. Remark 3. When g = (1,, 1) T, (3.10) and (3.15) share the same optima vaue. 3.3 A 0 Perturbed Hoisticay With amost the same argument of getting (2.9), the perturbed probem is min c T 0 x s.t. A i 0x + U i x b i 0, i = 1,, m, (3.16) x 0. 17

18 Then (1.7) for Mode 2 is formuated as s.t. A i 0x + U i x b i 0, i = 1,, m, (3.17) x 0, 0. It is interesting that the probem above has amost the same formuation of (3.10) by fixing g i = 1, i = 1, 2,..., m. When a the rows to be perturbed are in a row-wise fashion, the numbers of every row s perturbation direction vectors are the same, and a the perturbation radii are required to be equa, (3.10) of A 0 perturbed row-wisey is the same as (3.17) of A 0 perturbed hoisticay. Henceforth, the case of A 0 perturbed hoisticay is actuay a speciaization of A 0 perturbed row-wisey and a the resuts for A 0 perturbed row-wisey are true for A 0 perturbed hoisticay. 4 Numerica Experiments We are to impement the agorithm for probem (3.10) with g = (1,, 1) T, this means we et the radii of a rows be the same. Actuay, whether a entries of g are equa has no effect on the computationa compexity of the probem. It is reaized by MATLAB_R2015b on a aptop with Inte Core i5, CPU 2.7GHz and 8G memory. CVX-64 (version 2.0) ( is used to sove a convex programming probems here. The cardinaity of index set P is given as 30% of n, and x is supposed to be a strict interior point with x j = 0, j P and x j = 1, j / P. A 0 and c 0 are uniformy distributed in [ 10, 10] m n and [ 10, 10] n respectivey. And b 0 A 0 x A 0 is uniformy distributed in [2, 10] n to ensure A 0 x < b 0 hods stricty. Here symbo A 0 means conditiona distribution. Besides, we impose a box constraint x [ 100, 100] n to make the feasibe region bounded. Under these initia settings, Assumption 1 is tenabe. The number n i of perturbation vectors for the i-th row is set in two eves, 25% and 50% of n + m. And perturbation vectors u i j, j = 1,, n i, i = 1,, m are independenty uniformy distributed in [ 0.5, 0.5] n or [ 1, 1] n. (m, n) is set to be (20, 20), (20, 100) and (80, 100), and the precision ɛ is We wi show the improvement from the ower bound (3.11) to the imum radius, and 18

19 infuence on CPU time of different combination of (m, n, pert, scae), where m is the number of constraints, n is the dimensiona number of variabes, pert is the perturbation ratio 25% or 50%, and scae is the perturbation scae of u i j uniformy distributed in [ 0.5, 0.5]n or [ 1, 1] n. Here we et numbers of perturbation vectors for a the rows be equa for simpicity. In every setting (m, n, pert, scae), we do 10 random instances repeatedy. And Ave and Std stand for the average vaue and the standard deviation of the 10 experiments. Instance (m 1, n 2, pert 3, scae 4 ) Tabe 1: Numerica Resuts Lower Bound Maximum Radius Improvement Ratio 5 CPU Time Ave Std Ave Std Ave Std Ave Std (20,20,0.25,0.5) (20,20,0.25,1) (20,20,0.5,0.5) (20,20,0.5,1) (20,100,0.25,0.5) (20,100,0.25,1) (20,100,0.5,0.5) (20,100,0.5,1) (80,100,0.25,0.5) (80,100,0.25,1) (80,100,0.5,0.5) (80,100,0.5,1) number of constraints 2 dimension of x 3 ratio of number of perturbation vectors for one row to m + n 4 scae of perturbation vectors 5 improvement ratio=(imum radius-ower bound)/ower bound For these numerica experiments in Tabe 1, infinite perturbation radius doesn t appear since it is very rare and specia. We can see that the CPU time increases as the scae of the probem (m, n) increases, and m pays a more important part than n. Besides, the more the number of perturbation vectors is, the more CPU time it costs whereas the scae of perturbation vectors basicay has no infuence on CPU time. As for the imum radius, it is mainy affected by the scae of perturbation vectors. It is easiy found that the imum radius on eve [ 0.5, 0.5] n is approximatey twice the imum radius on eve [ 1, 1] n. And the other three factors have ess effect on it. And the improvement ratio is about between 2 and 6, which shows that a naive ower bound is too weak. 19

20 5 Notes At the end of the paper, we make two notes on why we haven t studied on the optima partition and another different formuation for Mode 2. First, the reason why we haven t studied on the optima partition, which is focused on in traditiona sensitivity anaysis for the inear programming, is that the perturbation probems no onger remain inear such as (2.5). And there is no point in keeping the optima partition unchanged, since the definition of optima partition of inear programming and second-order cone programming is different, see [13]. Especiay, for perturbation of b 0, (2.2) is uckiy a inear program and it can be regarded as a parametric inear programming probem, in which v is a given perturbation vector and is a parameter causing the variation. This probem has been studied detaiedy in [3], in which Ben-Ta and Nemirovski presented agorithms to compute a transition-points and showed that optima partition remains constant between two consecutive transition-points. Therefore the imum perturbation radius to keep the optima partition of (2.2) unchanged can be obtained by the methods mentioned in that paper. In order to estabish a unified anaysis system for those three cases, we consider Mode 2 to keep 0 entries remained, which is somehow ike the invariance of the optima partition. Second, if we consider the condition x j = 0, j P out of OPT 2 () for Mode 2, we obtain a new mode repacing (1.7) as foows: s.t. x OPT 1 (), 0. (5.1) For the genera case reformuation (3.10) of (1.7), we use the binary search agorithm to get the imum radii. When the objective of (5.1) is one dimensiona, which is simiar to that of (3.10), can we use the binary search agorithm to sove it? Here we consider the case that perturbation ony occurs in the right-hand-side vector as an exampe. According to the strong duaity theorem for the inear programming, (5.1) is 20

21 equivaent to the foowing biinear probem, s.t. ( A 0 I ) x = b 0 v, t A T 0 c y + s = 0, I 0 c T 0 x = (b 0 v) T y, x 0, t 0, s 0, 0, y R m. (5.2) For a fixed in (5.2), a constraints are inear. Then whether is feasibe for (5.2) can be determined in poynomia time. When 1 > 0 and 2 > 0 are both feasibe for (5.2), is feasibe for any ( 1, 2 )? If yes, we can use the binary search agorithm. Unfortunatey, OPT 1 () does not intersect the hyperpane x i = 0, i P for some. Then the binary search agorithm is invaid here. We present the foowing exampe to iustrate it. Exampe 2. min 30x 1 60x 2 20x 3 s.t. 3x 1 + 4x 2 + x 3 20, x 1 + 3x 2 + 2x 3 10, (5.3) x i 0, i = 1, 2, 3. The optima soution is x = (4, 2, 0) T and the perturbed vector for b is seected as v = (2, 0) T here. Let P = {3}. For fixed = 3, (0.4, 3.2, 0) T which is feasibe for (5.2) is an optima soution to (2.2). For fixed = 6, (5.2) is not feasibe and there exists no optima soution to (2.2) with the third entry being 0. 21

22 For = 10, (0, 0, 0) T is feasibe for (5.2) and is an optima soution to (2.2). The feasibe soution set of (2.2) is just a singeton set incuding (0, 0, 0) T. Thus = 3 and = 10 are both feasibe for (5.1) whie = 6 is not. That is why we consider mode (1.7) instead of (5.1) in our study. 6 Concusions In this study, we introduce a new concept robust sensitivity anaysis, which is reay different from the cassica sensitivity anaysis and some new variants of robust optimization. We have provided anayses of three cases in two scenarios. A modes in our study are poynomia computabe except for the ones containing more than one row perturbation in the constraint matrix. For those, we have deveoped a reativey easier binary search agorithm to get the imum perturbation radii and whether there exist poynomia modes with regard to them is sti an open probem. Furthermore, this concept can aso be used for the convex quadraticay constrained quadratica programming(cqcqp) probem, which is considered in another paper. Aso, the robust sensitivity anaysis for the semidefinite programming(sdp) probem is a potentia research direction. Acknowedgments This research has been supported by Nationa Natura Science Foundation of China Grant # and # Reference [1] A. Ben-Ta and A. Nemirovski. Robust soutions of uncertain inear programs. Operations Research Letters, 25(1):1 13, [2] Aharon Ben-Ta and Arkadi Nemirovski. Lectures on modern convex optimization. Society for Industria and Appied Mathematics, [3] Arjan B. Berkeaar, Kees Roos, and TamÃąs Teraky. The Optima Set and Optima Partition Approach to Linear and Quadratic Programming. Springer US, [4] Dimitris Bertsimas and David B. Brown. Constructing uncertainty sets for robust inear optimization. Operations Research, 57(6): ,

23 [5] Dimitris Bertsimas and Mevyn Sim. The price of robustness. Operations Research, 52(1):35 53, [6] Fiona Chandra, Dennice F. Gayme, Lijun Chen, and John C. Doye. Robustness, optimization, and architectures. European Journa of Contro, 17(5-6): , [7] George B. Dantzig. Linear Programming Under Uncertainty. INFORMS, [8] M. A. H Dempster. Introduction to stochastic programming. Springer New York, [9] Geir E Duerud and Fernando G Paganini. A Course in Robust Contro Theory a convex approach. Springer, New York, United States, [10] Ching Lai Hwang and Abu Syed Masud. Mutipe objective decision making-methods and appications: a state-of-the-art survey. Springer-Verag Berin Heideberg, [11] G. Infanger. Panning Under Uncertainty: Soving Large-Scae Stochastic Linear Programs. Linear Programming, [12] A. L. Soyster. Convex programming with set-incusive constraints and appications to inexact inear programming. Operations Research, 21(5): , [13] TamÃąs Teraky and Zhouhong Wang. On the identification of the optima partition of second order cone optimization probems. Siam Journa on Optimization, 24(1): , [14] Mi Ching Tsai and Da Wei Gu. Robust and optima contro. Automatica, 33(11): (1), [15] Guangin Xu and Samue Burer. Robust sensitivity anaysis of the optima vaue of inear programming. Optimization Methods and Software, onine. 23