The Optimal Set and Optimal Partition Approach to Linear. Arjan B. Berkelaar, Kees Roos and Tamças Terlaky. October 16, 1996

Transcription

1 The Optimal Set and Optimal Partition Approach to Linear and Quadratic Programming æ Arjan B. Berkelaar, Kees Roos and Tamças Terlaky October 16, 1996 Department of Econometrics and Operations Research Econometric Institute Faculty of Economics Erasmus University Rotterdam P.O.Box 1738, 3000 DR Rotterdam, The Netherlands. Abstract In this chapter we describe the optimal set approach for sensitivity analysis for LP.We show that optimal partitions and optimal sets remain constant between two consecutive transition-points of the optimal value function. The advantage of using this approach instead of the classical approach èusing optimal basesè is shown. Moreover, we present an algorithm to compute the partitions, optimal sets and the optimal value function. This is a new algorithm and uses primal and dual optimal solutions. We also extend some of the results to parametric quadratic programming, and discuss diæerences and resemblances with the linear programming case. Key words: Sensitivity Analysis, Parametric Programming, Complementarity æ This paper will appear as Chapter 6 in H. Greenberg and T. Gal èeditorsè, Recent Advances in Sensitivity Analysis and Parametric Programming, Kluwer Publishers, è1997è. 1

2 2 Optimal Set and Optimal Partition Approach 1 Introduction In this chapter we deal with parametric versions of linear programming èlpè and convex quadratic programming èqpè. First consider the LP problem èp è in standard format èp è min æ c T x : Ax = b; x ç 0 æ ; where c; x 2 IR n, b 2 IR m, and A is an m æ n matrix with full row rank. The dual problem èdè is written as èdè max æ b T y : A T y + s = c; s ç 0 æ ; where y 2 IR m and s 2 IR n. The input data for both problems consists of the matrix A, and the vectors b and c. In Chapter 4 it is shown that diæculties may arise, when the problem under consideration is degenerate. It is stated that the optimal basis may not be unique, that multiple optimal solutions may exist, and that the notion of shadow price is not correctly deæned 1. In Chapter 3 strictly complementary solutions are already mentioned. These solutions, although already shown to exist in 1956 by Goldman and Tucker ë10ë, came into view in the 1980s due to the immense popularity of interior point methods èden Hertog ë14ë and Jansen ë15ëè. This popularity was initiated by the seminal paper of Karmarkar ë17ë. Gíuler and Ye ë12ë showed that interior point methods for LP generate a strictly complementary solution èin the limitè. This triggered interest in investigating parametric analysis without using bases, instead making use of strictly complementary solutions èadler and Monteiro ë1ë, Mehrotra and Monteiro ë22ë, and Jansen et al. ë16ëè. In this chapter we describe this approach and formulate a new algorithm to compute the optimal value function, which uses primal and dual optimal solutions. Let us now turn to the quadratic programming formulation. The general CQP problem is given by ç èqp è min c T x + 1 ç 2 xt Qx : Ax = b; ç 0 ; where c; x 2 IR n, b 2 IR m, A an m æ n matrix with full row rank and Q a symmetric positive semideænite n æ n matrix. The Wolfeídual of èqp è is given by ç èqdè max b T y, 1 ç 2 ut Qu : A T y + s, Qu = c; s ç 0 ; where y 2 IR m and u; s 2 IR n. The input data for both problems consists of the matrix Q, the matrix A, and the vectors b and c. We assume that Q is positive semideænite resulting in a convex quadratic programming problem. Unless stated otherwise QP refers to convex quadratic programming problem. It is well known that if there exist optimal solutions for èqpè, then there also exist optimal solutions for which x = u. Furthermore, it is clear that LP is a special case of QP 2. In this chapter we are only concerned with changes in the vectors b and c, A and Q are taken to be æxed. Although sensitivity analysis and parametric programming for QP are not being performed on a large scale, there is at least one important application on a commercial level. The well-known Markowitz meanívariance model ë20ë for modern portfolio theory is formulated as a parametric QP. The optimal value function of this parametric QP is known as the eæcient frontier. The eæcient frontier is a useful tool, being used by various diæerent ænancial institutions for portfolio decision problems. Recent studies on computing the eæcient frontier èsee, e.g. ë19, 26, 27ëè all use optimal bases. Similar diæculties w.r.t. degeneracy as in the LP case exist for QP èthe optimal basis 3 1 See also, Gal ë9ë and Greenberg ë11ë for a survey and Rubin and Wagner ë24ë for an overview of practical implications. 2 Take Q to be the zero matrix. 3 Note that the primal or the dual QP problem may have a solution that is not a basic solution. However if we consider both the primal and the dual QP problem together, then, if the problem has a solution there always exists an optimal basis èrewrite the primal and dual problem as a linear complementarity problem to see thisè. In this chapter we take the freedom to speak about an optimal basis in the latter sense.

3 Optimal Set and Optimal Partition Approach 3 may not be unique, multiple solutions may exist, etc.è. Berkelaar et al. ë4ë consider the eæcient frontier as the outcome of their analysis for parametric QP using maximal complementary solutions. Interior point methods for QP generate such a maximal complementary solution èin the limitè. A related result to this was already shown by McLinden ë21ë in 1980 èsee also Gíuler and Ye ë13ëè. We describe parametric analysis for QP using maximal complementary solutions in this chapter, and again formulate an algorithm to compute the optimal value function. This algorithm uses both primal and dual solutions and their supports. Let us denote the optimal value of èqp è and èqdèaszèb; cè, with zèb; cè =,1 if èqp èisunbounded and èqdè infeasible and zèb; cè =1if èqdè isunbounded and èqp è infeasible. If èqp è and èqdè are both infeasible then zèb; cè is undeæned. We call z the optimal value function for the data b and c. Since LP is a special case of QP, we also denote the optimal value function for LP problems by zèb; cè with the same conventions. Although in the literature assumptions are often made to prevent situations concerning degeneracy, we shall not do so here. The main tools we use are the existence of strictly complementary and maximal complementary solutions in LP and QP respectively. Such solutions uniquely deæne the partition of the problem. We show that the pieces of the optimal value function correspond to intervals for the parameter on which the partition is constant. The proposed algorithms to compute the optimal value function are based on this key result. This chapter is organized as follows. In Section 2 we consider a transportation èlpè example to show that the classical approach to sensitivity analysis based on optimal bases leads to diæculties in case of degeneracy. Section 3 describes the optimal partition and we show how this concept and given optimal solutions can be used to characterize the optimal sets of LP and QP problems. In Section 4 we consider parametric LP based on the optimal set approach. We omit proofs in Section 4, and postpone them to Section 5. In Section 5 we consider parametric QP. Since LP is a special case of QP it is left to the reader to specialize the proofs to LP. Some results for LP and their proofs can be formulated and presented diæerently or need no proof at all. Section 4 has been organized so as to concentrate on the resemblances between parametric LP and parametric QP. For a more detailed analysis in LP the reader is referred to Jansen et al. ë15, 16ë. Finally we close this chapter by outlining how the ideas of Sections 4 and 5 can be applied to sensitivity analysis. 2 The Optimal Bases Approach - An Example In commercial packages for LP and QP usually the possibility is oæered to perform sensitivity analysis. As far as we know sensitivity analysis in all existing commercial packages is based on optimal bases. As a result, the outcome of the sensitivity analysis is often only partly correct. In this section we show this using an example. The classical approach to sensitivity analysis is based on pivoting methods èsuch as the Simplex method for LPè for solving LP and QP problems. These methods produce a so-called basic solution of the problem. It suæces for our purpose to know that such a solution is determined by anoptimal basis. We only brieæy consider a small textbook LP problem to illustrate problems that can occur in case of degeneracy. This example is taken from Jansen ë15ë. For a more detailed description we refer to Jansen et al. ë16ë and Jansen ë15ë. To illustrate the shortcomings of the implemented sensitivity analysis techniques we apply several commercial packages to a small LP problem. 2.1 Comparison of the classical and the new approach Example 1 We consider a simple transportation problem with three supply and demand nodes.

4 4 Optimal Set and Optimal Partition Approach Table 1: Sensitivity analysis for a transportation problem - RHS changes Ranges of supply and demand values LP-package b 1è2è b 2è6è b 3è5è b 4è3è b 5è3è b 6è3è CPLEX ë0,3ë ë4,7ë ë1,1è ë2,7ë ë2,5ë ë2,5ë LINDO ë1,3ë ë2,1è ë4,7ë ë2,4ë ë1,4ë ë1,7ë PC-PROG ë0,1è ë4,1è ë3,6ë ë2,5ë ë0,5ë ë2,5ë XMP ë0,3ë ë6,7ë ë1,1è ë2,3ë ë2,3ë ë2,7ë OSL ë0,3ë ë4,7ë è,1; 1è ë2,7ë ë2,5ë ë2,5ë Correct range ë0,1è ë2,1è ë1,1è ë0,7ë ë0,7ë ë0,7ë min P 9 i=1 x i s.t. x 1 + x 2 + x 3 + x 10 = 2 x 4 + x 5 + x 6 + x 11 = 6 x 7 + x 8 + x 9 + x 12 = 5 x 1 + x 4 + x 7, x 13 = 3 x 2 + x 5 + x 8, x 14 = 3 x 3 + x 6 + x 9, x 15 = 3 x i ç 0;i=1;:::;15. The results of a sensitivity analysis are shown in Table 1. The columns correspond to the RHS elements. The rows in the table correspond to æve packages CPLEX, LINDO, PC-PROG, XMP and OSL and show the ranges produced by these packages. The last row contains the ranges calculated by the approach outlined in this chapter 4. The diæerent ranges in the Table 1 are due to the diæerent optimal bases that are found by the diæerent packages. For each optimal basis the range can be calculated by examining for which values of the RHS element the optimal basis remains constant. The table demonstrates the weaknesses of the optimal bases approach that is implemented in the commercial packages. Sensitivity analysis is considered to be a tool for obtaining information about the bottlenecks and degrees of freedom in the problem. The information provided by the commercial packages is confusing and hardly allows a solid interpretation. The diæculties lie in the fact that an optimal basis need not be unique. 3 In this chapter we show that the optimal partition of a strictly complementary solution for LP or a maximal complementary solution for QP leads to a much more solid analysis. The reason for this is that the optimal partition is unique for any strictly or maximal complementary solution. 3 Optimal Partitions and Optimal Sets 3.1 Linear Programming The feasible regions of èp è and èdè are denoted as P := fx : Ax = b; x ç 0g ; D := æ èy; sè : A T y + s = c; s ç 0 æ : 4 The range provided by the IBM package OSL èoptimization Subroutine Libraryè for b 3 is not a subrange of the correct range; this must be due to a bug in OSL. The correct range for the optimal basis found by OSL should be ë1; 1è.

5 Optimal Set and Optimal Partition Approach 5 Assuming that èp è and èdè are both feasible, the optimal sets of èp è and èdè are denoted as P æ and D æ.we deæne the index sets B and N by B := fi : x i é 0 for some x 2P æ g; N := fi : s i é 0 for some èy; sè 2D æ g: The Duality Theorem for LP implies that B ë N = o=, and the Goldman and Tucker Theorem ë10ë that B ë N = f1; 2; æææ;ng: So, B and N form a partition of the full index set. This èorderedè partition, denoted as ç =èb; Nè, is the optimal partition of the problem èp è and of the problem èdè. In the rest of this chapter we assume that b and c are such that èp è and èdè have optimal solutions, and ç = èb; Nè denotes the optimal partition of both problems. By deænition, the optimal partition is determined by the set of optimal solutions for èp è and èdè. In this section it is made clear that, conversely, the optimal partition provides essential information on the optimal solution sets P æ and D æ. We use the notation x B and x N to refer to the restriction of a vector x 2 IR n to the coordinate sets B and N respectively. Similarly, A B denotes the restriction of A to the columns in B, and A N the restriction of A to the columns in N. Now the sets P æ and D æ can be described in terms of the optimal partition. The next lemma follows immediately from the Duality Theorem and the deænition of the optimal partition for LP and is therefore stated without proof. Lemma 1 Let x æ 2P æ and èy æ ;s æ è2d æ. Given the optimal partition èb; Nè ofèpè and èdè, the optimal sets of both problems are given by P æ = æ x : x 2P;x T s æ =0 æ = fx : x2p;x N =0g; D æ = æ èy; sè : èy; sè 2D;s T x æ =0 æ = fèy; sè : èy; sè 2D;s B =0g: The next result deals with the dimensions of the optimal sets of èp è and èdè. Here, as usual the èaæneè dimension of a subset of IR k is the dimension of the smallest aæne subspace in IR k containing the subset. Lemma 2 One has dimp æ = jbj,rank èa B è dimd æ = m, rank èa B è Lemma 2 immediately implies that èp è has a unique solution 5 if and only if rank èa B è=jbj. Clearly this happens if and only if the columns in A B are linearly independent. Also, èdè has a unique solution if and only if rank èa B è=m, which happens if and only if the rows in A B are linearly independent. Thus, both èpè and èdè have a unique solution if and only if A B is a basis èthe unique optimal basisè. 5 Notice that we speak of uniqueness of the optimal solution and not about optimal basis èwhich is not necessarily uniqueè.

6 6 Optimal Set and Optimal Partition Approach 3.2 Quadratic Programming In this section we describe analogous results, as in the preceding section, for QP. The proofs of these results are included here and it is left to the reader to specialize these proofs to LP. The feasible regions of èqp è and èqdè are denoted as QP := fx : Ax = b; x ç 0g ; QD := æ èu; y; sè : A T y +s,qu = c; s ç 0 æ : We start with the Duality Theorem for QP, which is stated without proof èsee, e.g. Dorn ë8ëè. Theorem 3 If x is feasible for èqp è and èu; y; sè for èqdè, then these solutions are optimal if and only if Qx = Qu and x T s =0. Assuming that èqp è and èqdè are both feasible, the optimal sets of èqp è and èqdè are denoted as QP æ and QD æ. These optimal sets can be characterized by maximal complementary solutions and the corresponding partition ègíuler and Ye ë13ëè. Let us deæne B := f i : x i é 0 for some x of QP æ g; N := f i : s i é 0 for some èu; y; sè ofqd æ g; T := f1;:::;ngnèbënè: The Duality Theorem for QP implies that B ën = o=. Note that, contrary to LP, in QP the Goldman and Tucker Theorem ë10ë does not hold, T may be nonempty. So B, Nand T form a partition of the full index set. This èorderedè partition, denoted as ç = èb; N; Tè, is the optimal partition of the problem èqp è and of the problem èqdè. A maximal complementary solution èx; y; sè is a solution for which x i é 0 èè i 2 B; s i é 0 èè i 2 N: The existence of such a solution is a consequence of the convexity of the optimal sets of èqp è and èqdè and was introduced by McLinden ë21ë. He showed an important result concerning such solutions, which was used by Gíuler and Ye ë13ë to show that interior point methods èas Anstreicher et al. ë2ë, Carpenter et al. ë5ë and Vanderbei ë25ëè generate such a solution èin the limitè. In Chapter 3 the deænition of the support of a vector was given. In this chapter we denote the support of a vector v by çèvè. Hence, given a strictly complementary or maximal complementary solution èx æ ;y æ ;s æ èwehaveb=çèx æ è and N = çès æ è. In the rest of this chapter we assume that b and c are such that èqp è and èqdè have optimal solutions, and ç = èb; N; Tè denote the optimal partition of both problems. By deænition, the optimal partition is determined by the set of optimal solutions for èqp è and èqdè. In this section we show that, conversely, the optimal partition provides essential information on the optimal solution sets QP æ and QD æ. In the introduction we already mentioned that there exist optimal solutions for which x = u; when useful we denote a dual optimal solution by èx; y; sè instead of èu; y; sè. Later we also use the following well-known result. Lemma 4 Let èx æ ; èu æ ;y æ ;s æ èè and è~x; è~u; ~y; ~sèè both be optimal solutions of èqp è and èqdè. Then Qx æ = Q~x = Qu æ = Q~u, c T x æ = c T ~x and b T y æ = b T ~y. Proof: Let us ærst consider the case where x æ = u æ and ~x =~u. Since èx æ ;y æ ;s æ è and è~x; ~y; ~sè are both optimal, we conclude that c T x æ xæ Qx æ = b T ~y, 1 2 ~xt Q~x:

7 Optimal Set and Optimal Partition Approach 7 Using that A T ~y +~s,q~x=cand multiplying by èx æ è T it follows, 1 2 èxæ Qx æ +~xq~x, 2~xQx æ è=b T ~y,c T x æ,~xqx æ =0: Thus, èx æ, ~xè T Qèx æ, ~xè = 0, which implies Qèx æ, ~xè = 0, i.e. Qx æ = Q~x, since Q is positive semideænite. Using that Qx æ = Q~x we conclude that c T x æ = c T ~x and b T y æ = b T ~y. From the Duality Theorem the proof is completed. 2 We use the notation x B, x N and x T to refer to the restriction of a vector x 2 IR n to the coordinate sets B, N and T respectively. Similarly, A B denotes the restriction of A to the columns in B, A N the restriction of A to the columns in N, and A T the restriction of A to the columns in T. The matrix Q is partitioned similarly. Now the sets QP æ and QD æ can be described in terms of the optimal partition. The next lemma follows immediately from the Duality Theorem and the deænition of the optimal partition for QP and is therefore stated without proof. Lemma 5 Let x æ 2QP æ and èu æ ;y æ ;s æ è 2QD æ. Given the optimal partition èb; N; Tè ofèqp è and èqdè, the optimal sets of both problems are given by QP æ = æ x : x 2QP;x T s æ =0;Qx=Qu ææ = fx : x 2QP;x NëT =0;Qx=Qu æ g ; QD æ = æ èu; y; sè : èu; y; sè 2QD;s T x æ =0;Qu=Qx ææ = fèu; y; sè : èu; y; sè 2QD;s BëT =0;Qu=Qx æ g : Lemma 6 Let M 2 IR næn be a symmetric positive semideænite matrix, partitioned according to B; N and T. Then N èm BB è=nèmæbè: Proof: Let x be an arbitrary vector in IR n and partitioned according to B and ç B = N ë T. Then x T Mx = x T B M BBx B +2x T ç B Mç BB x B + x T ç B M ç B ç B x ç B : The result is proven by contradiction. Suppose to the contrary that N èm BB è 6= NèMæBè. Take x B from the null space of M BB. Then, M BB x B = 0 and M ç BB x B 6= 0. Let " be given and consider x = "çx. Furthermore let æ = x T ç B è2m ç BB x B èè and æ = x T ç B M ç B ç B x ç B. It can easily be seen that æ 6= 0 and æ ç 0. Now we can rewrite x T Mx as follows x T Mx = "èx T ç B è2m ç BB çx B èè + " 2 èçx T ç B M ç B ç B çx ç B è="æ + " 2 æ: Thus, x T Mx é 0 if and only if "é,æ æ. Since M is positive semideænite we have a contradiction. This implies the result. 2 The next result deals with the dimensions of the optimal sets of èqp è and èqdè. Lemma 7 One has ç dim QP æ = jbj,rank dimqd æ = m, rank ç A T B A T T AB Q BB ç ç + n, rank èqè:

8 8 Optimal Set and Optimal Partition Approach Proof: given by Let a dual optimal solution u be given. Then, by Lemma 5 the optimal set of èqp èis QP æ = fx : Ax = b; x B ç 0; x NëT =0;Qx=Qug ; and hence the smallest aæne subspace of IR n containing QP æ is given by fx : A B x B = b; x NëT =0;QæBx B =Qug : The dimension of this aæne space is equal to the dimension of the null space of èa T B QT æb èt. Since the dimension of the null space of this matrix is given by jbj,rank èèa T B QT æb èt è, the ærst statement follows from Lemma 6. For the proof of the second statement we use that the dual optimal set can be described as in Lemma 5. Let x be a given primal solution. Then the optimal dual set is given by This is equivalent to QD æ = æ èu; y; sè : A T y +s,qu = c; s BëT =0;s N ç0;qu=qx æ : QD æ = fèu; y; sè : A T B y, èquè B = c B ; The smallest aæne subspace containing this set is A T T y, èquè T = c T ; A T N y + s N, èquè N = c N ; s B =0;s T =0;s N ç0;qu=qxg : QD æ = fèu; y; sè : A T B y, èquè B = c B ; A T T y, èquè T = c T ; A T N y + s N, èquè N = c N ; s B =0;s T =0;Qu=Qxg : Obviously s N is uniquely determined by y, and any y satisfying A T B y,èquè B = c B ;A T T y,èquè T = c T yields a point in the aæne èy; sè-space. Hence the dimension of this aæne space is equal to the dimension of the null space of èa B A T è T. The dimension of the null space of this matrix equals m, rank èèa B A T è T è. Furthermore, any u satisfying Qu = Qx yields a point in the aæne u-space. Hence the dimension of this aæne space is equal to the dimension of the null space of Q, which equals n, rank èqè. Combining these results completes the proof. 2 Note that when Q = 0 is substituted in the formulae of Lemma 7 the dimension of the dual set is not equal to the dimension in Lemma 2. Nevertheless, the results are consistent since u is not to be taken into account in the formula for the dimension of the dual optimal set. In this section we havecharacterized the optimal sets in LP and QP by using the optimal partition. In LP there is a one to one relation between the optimal partition and the primal and dual optimal sets. Given an optimal partition for an LP problem, the primal and dual optimal sets can be characterized. In QP we need the optimal partition and a dual optimal solution to characterize the primal optimal set; and the optimal partition and a primal optimal solution to characterize the dual optimal set. The results for parametric LP are thus obtained from parametric QP using the characterization of the optimal sets by the optimal partition. 4 Parametric Linear Programming In this section we investigate the eæects of changes in the vectors b and c on the optimal value zèb; cè of èp è and èdè. We consider one-dimensional parametric perturbations of b and c. So we want to study zèb + çæb; c + çæcè;

9 Optimal Set and Optimal Partition Approach 9 as a function of the parameters ç and ç, where æb and æc are given perturbation vectors. So, the vectors b and c are æxed, and the variations come from the parameters ç and ç. We denote the perturbed problems as èp ç è and èd ç è, and their feasible regions as P ç and D ç respectively. The dual problem of èp ç è is denoted as èd ç è and the dual problem of èd ç è is denoted as èp ç è. Observe that the feasible region of èd ç è is simply D and the feasible region of èp ç è is simply P. We use the superscript æ to refer to the optimal set of each of these problems. As we already mentioned in the introduction we postpone proofs that have an analogue in QP to the next section, where we consider parametric convex quadratic programming. We assume that b and c are such that èp è and èdè are both feasible. Then zèb; cè iswell deæned and ænite. It is convenient to introduce the following notations: bèçè :=b+çæb; cèçè :=c+çæc; fèçè :=zèbèçè; cè; gèçè:=zèb; cèçèè: Here the domain of the parameters ç and ç is taken as large as possible. Let us consider the domain of f. The function f is deæned as long as zèbèçè;cèiswell deæned. Since the feasible region of èd ç è is constant when ç varies, and since we assumed that èd ç è is feasible for ç = 0, it follows that èd ç è is feasible for all values of ç. Therefore, since fèçè iswell deæned if the dual problem èd ç è has an optimal solution and fèçè is not deæned èor inænityè if the dual problem èd ç èisunbounded. Using the Duality Theorem it follows that fèçè iswell deæned if and only if the primal problem èp ç èis feasible. In exactly the same way it can be understood that the domain of g consists of all ç for which èd ç è is feasible èand èp ç è boundedè. The following theorem is well known. Theorem 8 The domains of f and g are closed intervals on the real line. 4.1 Optimal value function and optimal sets on a linearity interval In this section we show that the functions fèçè and gèçè are piecewise linear on their domains. The pieces correspond to intervals where the partition is constant. For any ç in the domain of f we denote the optimal set of èp ç èasp æ ç and the optimal set of èd çèasd æ ç. The results in this section are related to similar results, already obtained in 1960s by e.g. Bereanu ë3ë, Charnes and Cooper ë6ë, and Kelly ë18ë èsee also Dinkelbach ë7, Chapter 5, Sections 1 and 2ë and Gal ë9ëè. The ærst theorem shows that the dual optimal set is constant on certain intervals and that f is linear on these intervals. This results from the fact that the optimal partition is constant on certain intervals èsee the next sectionè and the characterization of the optimal sets èsee Lemma 1è. Theorem 9 Let ç 1 and ç 2 éç 1 be such that D æ ç 1 = D æ ç 2. Then D æ ç is constant for all ç 2 ëç 1;ç 2 ë and fèçè is linear on the interval ëç 1 ;ç 2 ë. From this theorem we conclude the following result giving a partition of the domain of f in intervals on which the dual optimal set remains constant. Theorem 10 The domain of f can be partitioned in a ænite set of subintervals such that the dual optimal set is constant on a subinterval. Using the former two theorems we conclude that f is convex and piecewise linear. Theorem 11 The optimal value function fèçè is continuous, convex and piecewise linear. The values of ç where the partition of the optimal value function fèçè changes are called transitionpoints of f, and any interval between two successive transition-points of f is called a linearity interval of f. In a similar way we deæne transition-points and linearity intervals for g. Each of the above results on fèçè has its analogue for gèçè.

10 10 Optimal Set and Optimal Partition Approach Theorem 12 Let ç 1 and ç 2 éç 1 be such that P æ ç 1 = P æ ç 2. Then P æ ç is constant for all ç 2 ëç 1 ;ç 2 ë and gèçè is linear on the interval ëç 1 ;ç 2 ë. Theorem 13 The domain of g can be partitioned in a ænite set of subintervals such that the primal optimal set is constant on a subinterval. Theorem 14 The optimal value function gèçè is continuous, concave and piecewise linear. 4.2 Extreme points of a linearity interval In this section we assume that ç belongs to the interior of a linearity interval ëç 1 ;ç 2 ë. Given an optimal solution of èdç ç èwe show how the extreme points ç 1 and ç 2 of the linearity interval containing ç can be found by solving two auxiliary linear optimization problems. This is stated in the next theorem. Theorem 15 Let ç ç be arbitrary and let èy æ ;s æ èbeany optimal solution of èdç ç è. Then the extreme points of the linearity interval ëç 1 ;ç 2 ë containing ç ç follow from æ ç 1 = min ç : Ax = b + çæb; x ç 0; x T s æ =0 æ ç;x æ ç 2 = max ç : Ax = b + çæb; x ç 0; x T s æ =0 æ : ç;x Theorem 16 Let ç ç be a transition-point and let èy æ ;s æ è be a strictly complementary optimal solution of èdç ç è. Then the numbers ç 1 and ç 2 given by Theorem 15 satisfy ç 1 = ç 2 = ç ç. Proof: If èy æ ;s æ è is a strictly complementary optimal solution of èdç ç è then it uniquely determines the optimal partition of èdç ç è and this partition diæers from the optimal partition corresponding to the optimal sets at the linearity intervals surrounding ç. ç Hence èy æ ;s æ è does not belong to the optimal sets at the linearity intervals surrounding ç. ç Furthermore it holds in any transition-point that that æb T y æ satisæes æb T y, é æb T y æ é æb T y + ; where y, belongs to the linearity interval just to the left of y and y + belongs to the linearity interval just to the right of y. Hence, the theorem follows. 2 The corresponding results for g are stated below. Theorem 17 Let ç be arbitrary and let x æ be any optimal solution of èp ç è. Then the extreme points of the linearity interval ëç 1 ;ç 2 ë containing ç follow from æ ç 1 = min ç : A T y + s = c + çæc; s ç 0; s T x æ =0 æ ç;y;s æ ç 2 = max ç : A T y + s = c + çæc; s ç 0; s T x æ =0 æ : ç;y;s Theorem 18 Let ç be a transition-point and let x æ be a strictly complementary optimal solution of èp ç è. Then the numbers ç 1 and ç 2 given by Theorem 17 satisæes ç 1 = ç 2 =ç.

11 Optimal Set and Optimal Partition Approach Left and right derivatives of the value function and optimal sets in a transitionpoint In this section we show that the transition-points occur exactly where the the optimal value function is not diæerentiable. We have already seen that the optimal set remains constant on linearity intervals. We ærst deal with the diæerentiability offèçè. If the domain of f has a right extreme point then we may consider the right derivative at this point to be 1, and if the domain of f has a left extreme point the left derivative at this point maybe taken,1. Then we maysay that ç is a transition-point off if and only if the left and the right derivative of f at ç are diæerent. This follows from the deænition of a transition-point. Denoting the left and the right derivative asf 0,èçè and f 0 +èçè respectively, the convexity offimplies that at a transition-point ç one has f 0,èçè éf 0 +èçè: If domèfè has a right extreme point then it is convenient to consider the open interval at the right of this point as a linearity interval where both f and its derivative are 1. Similarly, if domèfè has a left extreme point then we may consider the open interval at the left of this point as a linearity interval where both f and its derivative are,1. Obviously, these extreme linearity intervals are characterized by the fact that on the intervals the primal problem is infeasible and the dual problem unbounded. The dual problem is unbounded if and only if the set D æ ç of optimal solutions is empty. Theorem 19 Let ç 2 domèfè and let x æ be any optimal solution of èp ç è. Then the derivatives at ç satisfy f,èçè 0 æ = min æb T y : A T y + s = c; s ç 0; s T x æ =0 æ y;s f+èçè 0 æ = max æb T y : A T y + s = c; s ç 0; s T x æ =0 æ : y;s Note that we could also have uses the deænition of the optimal set D æ ç in the above theorem. The above theorem reveals that æb T y must have the same value for all y 2D æ ç and for all ç 2 èç 1;ç 2 è. So we may state Corollary 20 Let ç 2 dom èfè belong to the linearity interval èç 1 ;ç 2 è. Then one has f 0 èçè =æb T y; 8ç 2 èç 1 ;ç 2 è; 8y 2D æ ç : By continuity we may write fèçè =b T çy+çæb T çy=bèçè T çy; 8ç 2 ëç 1 ;ç 2 ë: Lemma 21 Let ç 2 dom èfè belong to the linearity interval èç 1 ;ç 2 è. Moreover, let D æ èç 1;ç 2è := Dæ ç for arbitrary ç 2 èç 1 ;ç 2 è. Then one has D æ èç 1;ç 2è çdæ ç 1 ; D æ èç 1;ç 2è çdæ ç 2 : Corollary 22 Let ç be a nonextreme transition-point off and let ç + belong to the open linearity interval just to the right ofçand ç, to the open linearity interval just to the left of ç. Then we have D æ ç, çdæ ç ; Dæ ç +çdæ ç ; Dæ ç,ëdæ ç + =o=; where the inclusions are strict. Two other almost obvious consequences of the above results are the following corollaries.

12 12 Optimal Set and Optimal Partition Approach Corollary 23 Let ç be a nonextreme transition-point offand let ç + and ç, be as deæned in Corollary 22. Then we have D æ ç,=æ y2d æ ç : æb T y =æb T y,æ ; D æ ç +=æ y2d æ ç : æb T y =æb T y +æ : Corollary 24 Let ç be a nonextreme transition-point offand let ç + and ç, be as deæned in Corollary 22. Then dimd æ ç, é dim Dæ ç ; dimdæ ç + é dimdæ ç : The picture becomes more complete now. Note that Theorem 19 is valid for any value of ç in the domain of f. The theorem re-establishes that at a `non-transition' point, where the left and right derivative offare equal, the value of æb T y is constant when y runs through the dual optimal set D æ ç. But it also makes clear that at a transition-point, where the two derivatives are diæerent, æbt y is not constant when y runs through the dual optimal set D æ ç. Then the extreme values of æbt y yield the left and the right derivative offat ç; the left derivative is the minimum and the right derivative the maximal value of æb T y when y runs through the dual optimal set D æ ç. The reverse of Theorem 9 also holds. This is stated in the following theorem. Theorem 25 If fèçè is linear on the interval ëç 1 ;ç 2 ë, where ç 1 éç 2, then the dual optimal set D æ ç is constant for ç 2 èç 1 ;ç 2 è. If ç ç is not a transition-point then there is only one linearity interval containing ç ç, and hence this must be the linearity interval ëç 1 ;ç 2 ë, as given by Theorem 15. It may beworthwhile to point out that if ç ç is a transition-point, however, there are three linearity intervals containing ç ç, namely the singleton interval ë ç ç; ç çë and the two surrounding linearity intervals. In that case, the linearity interval ëç 1 ;ç 2 ë given by Theorem 15 may beany of these three intervals, and which of the three intervals is gotten depends on the given optimal solution èy æ ;s æ èofèdç ç è. It can easily be understood that the linearity interval at the right of ç çis found if èy æ ;s æ è happens to be optimal on the right linearity interval. This occurs exactly when æb T y æ = f 0 +è ç çè, due to Corollary 23. Similarly, the linearity interval at the left of ç ç is found if èy æ ;s æ è is optimal on the left linearity interval and this occurs exactly when æb T y æ = f 0,è ç çè, also due to Corollary 23. Finally, if è1è f 0,è ç çèéæb T y æ éf 0 +è ç çè; then we haveç 1 =ç 2 = ç çin Theorem 15. The last situation seems to be most informative. It clearly indicates that ç ç is a transition-point off, which is not apparent in the other two situations. Knowing that ç ç is a transition-point off we can ænd the two one-sided derivatives of f at ç ç as well as optimal solutions for the two surrounding interval at ç ç from Theorem 19. Remark 1 It is interesting to consider the dual optimal set D æ ç when ç runs from,1 to 1. Left from the smallest transition-point èthe transition-point for which ç is minimalè the set D æ ç is constant. It may happen that D æ ç is empty there, due to the absence of optimal solutions for these small values of ç. This occurs if èd ç èisunbounded èwhich means that èp ç è is infeasibleè for the values of ç on the most left open linearity interval. Then, at the ærst transition-point, the set D æ ç increases to a larger set, and when passing to the next open linearity interval the set D æ ç becomes equal to a proper subset of this enlarged set. This process repeats itself at every new transitionpoint: at a transition-point offthe dual optimal set expends itself and when passing to the next open linearity interval it shrinks to a proper subset of the enlarged set. Since the derivative off is monotonically increasing when ç runs from,1 to 1 every new dual optimal set arising in this way diæers from all previous ones. In other words, every transition-point off and every linearity interval of f has its own dual optimal set.

13 Optimal Set and Optimal Partition Approach 13 Each of the above results has a dual analogy for gèçè. Lemma 26 Let ç belong to the interior of dom ègè and let ç + belong to the open linearity interval just to the right ofçand ç, to the open linearity interval just to the left of ç. Moreover, let x + 2P æ ç and x, 2P æ + ç. Then one has, g,èçè 0 æ = max æc T x : x 2P æ x çæ =æc T x, g+èçè 0 æ = min æc T x : x 2P æ æ ç =æc T x + : x Theorem 27 Let ç 2 domègè and let èy æ ;s æ èbeany optimal solution of èd ç è. Then the derivatives at ç satisfy g, 0 æ èçè = max æc T x : Ax = b; x ç 0; x T s æ =0 æ x g+èçè 0 æ = min æc T x : Ax = b; x ç 0; x T s æ =0 æ : x Corollary 28 Under the hypothesis of Theorem 25 one has g 0 èçè =æc T x; 8ç 2 èç 1 ;ç 2 è; 8x2P æ ç : Corollary 29 Let ç 2 dom ègè belong to the linearityinterval èç 1 ;ç 2 è. Moreover, let P æ èç 1;ç 2è := Pæ ç for arbitrary ç 2 èç 1 ;ç 2 è. Then one has P æ èç 1;ç 2è çpæ ç 1 ; P æ èç 1;ç 2è çpæ ç 2 : Corollary 30 Let ç be a nonextreme transition-point ofgand let ç + belong to the open linearity interval just to the right ofçand ç, to the open linearity interval just to the left of ç. Then we have P æ ç, çpæ ç ; Pæ ç +çpæ ç ; Pæ ç,ëpæ ç + =o=; where the inclusions are strict. Corollary 31 Let ç be a nonextreme transition-point ofgand let ç + and ç, be as deæned in Corollary 30. Then we have P æ ç,=æ x2p æ ç : æc T x =æb T x,æ ; P æ ç +=æ x2p æ ç : æc T x =æb T x +æ : Corollary 32 Let ç be a nonextreme transition-point ofgand let ç + and ç, be as deæned in Corollary 30. Then dimp æ ç, é dimpæ ç ; dimpæ ç + é dimpæ ç : Theorem 33 If gèçè is linear on the interval ëç 1 ;ç 2 ë, where ç 1 éç 2, then the primal optimal set P æ ç is constant for ç 2 èç 1 ;ç 2 è. 4.4 Computing the optimal value function Using the results of the previous sections, we present in this section an algorithm which yields the optimal value function for a one-dimensional perturbation of the vector b or the vector c. We ærst deal with a one-dimensional perturbation of the vector b with a scalar multiple of the vector æb; we state the algorithm for the calculation of the optimal value function and that the algorithm ænds all the transition-points and linearity intervals of it. Having done this it is clear how to treat

14 14 Optimal Set and Optimal Partition Approach a one-dimensional perturbation of the vector c; we also state the corresponding algorithm and its convergence results. Assume that we have given optimal solutions x æ of èp è and èy æ ;s æ èofèdè. Using the notation of the previous sections, the problems èp ç è and its dual èd ç è arise by replacing the vector b by bèçè = b+çæb; the optimal value of these problems is denoted as fèçè. So wehavefè0è = c T x æ = b T y æ. The domain of the optimal value function is è,1; 1è and fèçè =1if and only if èd ç èisunbounded. Recall from Theorem 11 that fèçè is convex and piecewise linear. Algorithm 1 determines f on the nonnegative part of the real line. We leave it to the reader to ænd some straightforward modiæcations of the algorithm yielding an algorithm which generates f on the other part of the real line. Input: An optimal solution x æ of èp è; An optimal solution èy æ ;s æ èofèdè; a perturbation vector æb. begin ready:=false; k := 1; x 0 := x æ ; y 0 := y æ ; s 0 = s æ ; Solve f 0 +è0è = max y;s æ æb T y : A T y + s = c; s ç 0; s T x 0 =0 æ ; while not ready do begin Solve max ç;x æ ç : Ax = b + çæb; x ç 0; x T s k,1 =0 æ ; if this problem is unbounded: ready:=true else let èç k ;x k è be an optimal solution; begin Solve f 0 +èç k è = max y;s æ æb T y : A T y + s = c; s ç 0; s T x k =0 æ ; if this problem is unbounded: ready:=true else let èy k ;s k è be an optimal solution; k := k +1; end end end Algorithm 1: Optimal Value Function fèçè; çç0 The following theorem states that Algorithm 1 ænds the successive transition-points of f on the nonnegative part of the real line, as well as the slopes of f on the successive linearity intervals. Theorem 34 Algorithm 1 terminates after a ænite number of iterations. If K is the number of iterations upon termination, then ç 1 ;ç 2 ;æææ;ç K are the successive transition-points of f on the nonnegative real line. The optimal value at ç k è1 ç k ç Kè isgiven by c T x k, and the slope of f on the interval èç k ;ç k+1 èè1çkékèbyæb T y k. From Algorithm 1 we ænd the linearity intervals and the slope of f on these intervals. The optimal value function can now easily be drawn with these ingredients. When perturbing the vector c with a scalar multiple of æc to cèçè =c+çæcthe algorithm for the calculation of the optimal value function gèçè can be stated as in Algorithm 2 èrecall that g is concaveè. Algorithm 2 ænds the successive transition-points of g on the nonnegative real line as well as the slopes of g on the successive linearity intervals.

15 Optimal Set and Optimal Partition Approach 15 Input: An optimal solution x æ of èp è; An optimal solution èy æ ;s æ èofèdè; a perturbation vector æc. begin ready:=false; k := 1; x 0 := x æ ; s 0 = s æ ; y 0 = y æ Solve g 0 +è0è = min x æ æc T x : Ax = b; x ç 0; x T s 0 =0 æ ; while not ready do begin Solve max ç;y;s æ ç : A T y + s = c + çæc; s ç 0; s T x k,1 =0 æ ; if this problem is unbounded: ready:=true else let èç k ;y k ;s k è be an optimal solution; begin Solve g 0 + èç kè = min x æ æc T x : Ax = b; x ç 0; x T s k =0 æ ; if this problem is unbounded: ready:=true else let x k be an optimal solution; k := k +1; end end end Algorithm 2: Optimal Value Function gèçè; çç0 Theorem 35 Algorithm 2 terminates after a ænite number of iterations. If K is the number of iterations upon termination, then ç 1 ;ç 2 ;æææ;ç K are the successive transition-points of g on the nonnegative real line. The optimal value at ç k è1 ç k ç Kè isgiven by b T y k, and the slope of g on the interval èç k ;ç k+1 èè1çkékèbyæc T x k. 5 Parametric Quadratic Programming In this section we start to investigate the eæect of changes in b and c on the optimal value zèb; cè of èqp è and èqdè. Again, as in LP, we consider one-dimensional parametric perturbations of b and c. So we want to study zèb + çæb; c + çæcè; as a function of the parameters ç and ç, where æb and æc are given perturbation vectors. So, again the vectors b and c are æxed, and the variations come from the parameters ç and ç. The perturbed problems are denoted as èqp ç è and èqd ç è, and their feasible regions as QP ç and QD ç respectively. The dual problem of èqp ç è is denoted as èqd ç è and the dual problem of èqd ç è is denoted as èqp ç è. Observe that the feasible region of èqd ç è is simply QD and the feasible region of èqp ç è is simply QP. Again, we use the superscript æ to refer to the optimal set of each of these problems. We assume that b and c are such that èqp è and èqdè are both feasible. Then, zèb; cè is again well deæned and ænite. We use the following notations again: bèçè :=b+çæb; cèçè :=c+çæc; fèçè :=zèbèçè; cè; gèçè:=zèb; cèçèè: The domain of the parameters ç and ç is again taken as large as possible. Let us consider the domain of f. This function is deæned as long as zèbèçè;cè is well deæned. Therefore, fèçè is well deæned if

16 16 Optimal Set and Optimal Partition Approach the dual problem èqd ç è has an optimal solution and fèçè is not deæned èor equals inænityè if the dual problem èqd ç èisunbounded. Using the Duality Theorem it follows that fèçè iswell deæned if and only if the primal problem èqp ç è is feasible. In exactly the same way it can be understood that the domain of g consists of all ç for which èqd ç è is feasible èhence èqp ç è boundedè. Note the new meaning of the functions f and g. We prefer to use the same notation as in the preceding section. However, these functions are deæned diæerently. Lemma 36 The domains of f and g are convex. Proof: We give the proof for f. The proof for g is similar and therefore omitted. Let ç 1 ;ç 2 2 domèfè and ç 1 éçéç 2. Then fèç 1 è and fèç 2 è are ænite, which means that both QP ç1 and QP ç2 are nonempty. Let x 1 2QP ç1 and x 2 2QP ç2. Then x 1 and x 2 are nonnegative and Ax 1 = b + ç 1 æb; Ax 2 = b + ç 2 æb: Now consider x := x 1 + ç, ç 1 ç 2, ç 1, x 2, x 1æ = èç 2, çè x 1 +èç,ç 1 èx 2 ç 2,ç 1 : Note that x is a convex combination of x 1 and x 2 and hence x is nonnegative. We proceed by showing that x 2QP ç. Using that A, x 2, x 1æ =èç 2,ç 1 èæb this goes as follows: Ax = Ax 1 + ç, ç 1 ç 2, ç 1 A, x 2, x 1æ = b + ç 1 æb + ç, ç 1 ç 2, ç 1 èç 2, ç 1 èæb = b+ç 1 æb+èç,ç 1 èæb = b+çæb: This proves that èqp ç è is feasible and hence ç 2 domèfè, completing the proof. 2 Lemma 37 The functions f and g are convex. Proof: We present the proof only for f, the proof for g is similar. Let ç 1 ;ç 2 be elements of the interior of the domain of f. Let æ 2 è0; 1è be given and deæne ç æ := æç 1 +è1,æèç 2. Then we have fèç æ è = æ çèb+ç 1 æbè T y èçæè, 12 ç èxèçæè è T Qx èçæè + è1, æè çèb + ç 2 æbè T y èçæè, 12 ç èxèçæè è T Qx èçæè ç æfèç 1 è+è1,æèfèç 2 è; where the inequality holds since the feasible set of èqd ç è is independent ofç. 2 Lemma 38 The complements of the domains of f and g are open intervals of the real line.

17 Optimal Set and Optimal Partition Approach 17 Proof: Let ç belong to the complement of the domain of f. This means that èqd ç èisunbounded. This is equivalent to the existence of a vector z, such that A T z ç 0; èb + çæbè T zé0: Fixing z and considering ç as a variable, the set of all ç satisfying èb + çæbè T z é0isanopen interval. For all ç in this interval èqd ç èisunbounded. Hence, the complement of the domain of f is open. This concludes the proof. 2 A consequence of the last two lemmas is the next theorem ècf. Theorem 8è which requires no further proof. Theorem 39 The domains of f and g are closed intervals on the real line. 5.1 Optimal value function and optimal partitions on a curvy-linearity interval In this section we show that the optimal value functions fèçè and gèçè are piecewise quadratic on their domains. The pieces correspond to intervals where the partition is constant. In LP these results are given in terms of the optimal primal and dual sets. In QP this is not possible since these sets are intertwined. Neither èqp ç è nor èqd ç è is constant when ç or ç vary. The proofs for LP are obtained by using the characterization of the optimal set by the optimal partition èsee Section 6.3è. For any ç in the domain of f we denote the optimal set of èqp ç èasqp æ ç and the optimal set of èqd ç èasqd æ ç. The ærst theorem ècf. Theorem 9è shows that the partition is constant on certain intervals and that f is quadratic on these intervals. Theorem 40 Let ç 1 and ç 2 éç 1 be such that ç ç1 = ç ç2. Then ç ç is constant for all ç 2 ëç 1 ;ç 2 ë and fèçè is quadratic on the interval ëç 1 ;ç 2 ë. Proof: Without loss of generalitywe assume that ç 1 = 0 and ç 2 = 1. Let èx èç1è ; èu èç1è ;y èç1è ;s èç1è èè and èx èç2è ; èu èç2è ;y èç2è ;s èç2è èè be maximal complementary solutions for the respective problems. We assume that u èç1è = x èç1è and u èç2è = x èç2è and deæne for ç 2 è0; 1è è2è xèçè :=è1,çèx èç1è +çx èç2è ; yèçè :=è1,çèy èç1è +çy èç2è ; sèçè :=è1,çès èç1è +çs èç2è : It is easy to see that Axèçè =b+çæband xèçè ç 0. Also A T yèçè+sèçè,qxèçè =è1,çèc+çc = c; and sèçè ç 0. So èxèçè;yèçè;sèçèè is feasible for èqp ç è and èqd ç è. Since ç 0 = ç 1,wehave xèçè T sèçè = 0, hence the proposed solution is optimal for èqp ç è and èqd ç è. Using the support of xèçè and sèçè, B and N respectively, and their complement T, this implies è3è B ç B ç ; N ç N ç and T ç T ç : We now show that equality holds. Assuming to the contrary that T ç T ç, there exists a maximal complementary solution èx èçè ;y èçè ;s èçè è of èqp ç è and èqd ç è such that è4è Let us now deæne for æé0 èx èçè è i +ès èçè è i é0 for some i 2 T; i =2 T ç : çxèæè:=x èç2è +æèx èç2è,x èç1è è; çyèæè:=y èç2è +æèy èç2è,y èç1è è; çsèæè:=s èç2è +æès èç2è,s èç1è è:

18 18 Optimal Set and Optimal Partition Approach For some æé0 small enough it holds è5è çxèæè B é 0; çxèæè NëT =0; çsèæè N é0; çsèæè BëT =0; from which it follows that the proposed solutions are optimal for èqp 1+æ è and èqd 1+æ è. Finally, we deæne ~x := æ 1,ç+æ xèçè + 1,ç 1,ç+æ çxèæè; which are feasible in èqp 1 è and èqd 1 è. Also, Since è3è and è5è imply and ~y := æ 1,ç+æ yèçè + 1,ç 1,ç+æ çyèæè; ~s := æ 1,ç+æ sèçè + 1,ç 1,ç+æ çsèæè; ~x T æ 1, ç ~s = 1, ç + æ 1, ç + æ ç èx èçè è T çsèæè+çxèæè T s èçèç : èx èçè è T çsèæè =èx èçè è T Nçsèæè N =0 ès èçè è T çxèæè =ès èçè è T Bçxèæè B =0; è~x; ~y; ~sè is optimal for èqp 1 è and èqd 1 è. However, if è4è would hold, we would have a solution of èqp 1 è and èqd 1 è with either è~xè i é 0orè~sè i é0 for i 2 T, contradicting the deænition of èb; N; Tè. Thus we conclude T ç = T. Using è3è the ærst part of the theorem follows. The second part can now be proven almost straightforwardly. From the proof of the ærst part we know that èxèçè;yèçè;sèçèè deæned in è2è is optimal in èqd ç è for ç 2 è0; 1è. Hence Note that fèçè = èb + çæbè T yèçè, 1 2 xèçèt Qxèçè = fè0è + çæb T y èç1è + çb T èy èç2è, y èç1è è+ç 2 æb T èy èç2è,y èç1è è, çèx èç2è,x èç1è è T Qx èç1è, 1 2 ç2 èx èç2è, x èç1è è T Qèx èç2è, x èç1è è: Aèx èç2è, x èç1è è = æb A T èy èç2è, y èç1è è+s èç2è,s èç1è = Qèx èç2è,x èç1è è: Multiplying the second equation with x èç2è, x èç1è respectively with x èç1è, and using the ærst, gives è6è è7è So we may write fèçè as æb T èy èç2è, y èç1è è = èx èç2è, x èç1è è T Qèx èç2è, x èç1è è b T èy èç2è, y èç1è è = èx èç1è è T Qèx èç2è, x èç1è è: fèçè=fè0è + çæb T y èç1è ç2 æb T èy èç2è, y èç1è è: This concludes the proof. 2 The function f on ëç 1 ;ç 2 ë is explicitly given by the following formula ç 2 fèçè = ç2 æç èbt y èç 1 è è, 1 2æç èxèç 1 è è T Qx èç 1 è + ç1 æç èbt y ç2 è, 1 2 æç èxèç 2 è è T Qx èç 2 è + ç æç ç c T èx èç 2 è, x èç 1 è è+è2,ç 1,ç 2èb T èy èç 2 è,y èç 1 è è+ 2ç2ç1 æç æbt èy èç 2 è,y èç 1 è è ç 2 æç æbt èy èç 2 è,y èç 1 è è: Note that we can now calculate the optimal value function between two subsequent transition-points. Theorem 40 implies the following corollary. ç 1 ç

19 Optimal Set and Optimal Partition Approach 19 Corollary 41 If ç ç1 = ç ç2 = ç then fèçè is linear on ëç 1 ;ç 2 ë if and only if Qx èç1è = Qx èç2è. Proof: Assuming again ç 1 = 0 and ç 2 =1,wehave from the proof of Theorem 40 that fèçè is linear on ë0; 1ë if and only if æb T èy èç2è, y èç1è è = 0. Using è6è this is equivalent to èx èç2è,x èç1è è T Qèx èç2è,x èç1è è=0 which holds if and only if Qèx èç2è, x èç1è è. 2 As a consequence of Theorem 40 we have the following results ècf. Theorem 9 and Theorem 11. Theorem 42 The domain of f can be partitioned in a ænite set of subintervals such that the optimal partition is constant on a subinterval. Proof: Since the number of possible partitions is ænite and the number of elements in the domain of f is inænite, it follows from Theorem 40 that the domain of f can be partitioned into èopenè subintervals on which the partition is constant, while it is diæerent in the singletons in between the subintervals. This implies the result. 2 Theorem 43 The optimal value function fèçè is continuous, convex and piecewise quadratic. Proof: Corollary 41 implies that on each subinterval deæned by a partition the function fèçè is quadratic. Since fèçè is convex èlemma 37è it is continuous on the interior of its domain. It remains to be shown that the optimal value function is right-continuous and left-continuous in the left and right endpoints of the domain of f respectively. To this end we consider the left endpoint of the domain of f èfor the right endpoint the proof is similarè. Let ç æ denote the left endpoint. Now, we need to proof that lim fèçè =fèçæ è: çèç æ Let èçx; çy; çsè denote the optimal solution at the left endpoint ç æ.furthermore consider the limit of èxèçè;yèçè;sèçèè for ç è ç æ. Since the dual feasible set èqd ç è is closed and is independent ofç, the limit point ~y; ~s of yèçè;sèçè is dual feasible. Further the limit point ~xof the sequence xèçè is feasible for èqp ç æè. Since the sequences are complementary, the limit points are also complementary, hence optimal. Applying Lemma 4 completes the proof. 2 The values of ç where the partition of the optimal value function fèçè changes are called transitionpoints of f, and any interval between two successive transition-points of f is called a curvy-linearity interval of f. In a similar way we deæne transition-points and curvy-linearity intervals for g. Each of the above results on fèçè has its analogue for gèçè. We state these results without further proof. The omitted proofs are straightforward modiæcations of the above proofs. Theorem 44 Let ç 1 and ç 2 éç 1 be such that ç ç1 = ç ç2. Then ç ç is constant for all ç 2 ëç 1 ;ç 2 ë and gèçè is quadratic on the interval ëç 1 ;ç 2 ë. Corollary 45 If ç ç1 = ç ç2 = ç then gèçè is linear on ëç 1 ;ç 2 ë if and only if Qx èç1è = Qx èç2è. Theorem 46 The domain of g can be partitioned in a ænite set of subintervals such that the optimal partition is constant on a subinterval.