ON THE UNIQUENESS OF THE OPTIMAL SOLUTION IN LINEAR PROGRAMMING

. In this paper numerous necessary and suﬃcient conditions will be given for a vector to be the unique optimal solution of the primal problem, as well as for that of the dual problem, and even for the case when the primal and the dual problem have unique optimal solutions at the same time, respectively, by means of using the strict complementarity and the linear independence constraint qualiﬁcation. Beyond that, the topological structure of the optimal solutions satisfying the strict complementarity will be determined.

There are a number of trivial and non-trivial examples for sufficient conditions as well as for necessary conditions for the uniqueness of the primal optimal solution, and also for those of the dual optimal solution.For example, it is obvious that if (P) has only one feasible point, it is, of course, an optimal solution, too; and this is also true for (D).Hence, the trivial condition that the corresponding feasible set has one element, is a sufficient condition for the uniqueness of the optimal solution for (P) and (D), respectively.From the strict complementary slackness theorem it follows that -if there is an optimal solution either for (P) or for (D) -there are optimal solutions of (P) and (D) that meet the strict complementary condition (SCC).Thus, this last condition is a necessary condition for the uniqueness of the optimal solution for (P) and (D), respectively.Some other examples.It is well-known that if there is an optimal solution x 0 for (P) and the linear independence constraint qualification (LICQ) is fulfilled at x 0 -i.e. the rows of A corresponding to the active indices at x 0 are linearly independent -then the dual has exactly one optimal solution.
(The set of active indices at a primal feasible solution x 0 is defined as while at a dual feasible solution y 0 is defined as I(y 0 ) = {i ∈ I; y 0 i = 0}.y 0 i denotes the ith element of vector y 0 .)Therefore, satisfaction of LICQ at a primal optimal solution x 0 is a sufficient condition for the uniqueness of the dual optimal solution.On the other hand, it is also known that if the primal feasible set is nonempty, but it has no inner point, then there is no unique dual optimal solution.Hence, the Slater constraint qualification is a necessary condition for the uniqueness of the dual optimal solution.The above-mentioned conditions are either necessary or sufficient conditions.However, it would be important to get necessary and sufficient conditions for the uniqueness • of the primal optimal solution • of the dual optimal solution • of both the primal and at the same time the dual optimal solution.
In the paper results of that kind will be presented as a primary purpose, with the help of LICQ and SCC.Although, in [3] some necessary and sufficient conditions for the uniqueness of the primal optimal solution have been presented, they are too difficult to apply.Thus, the secondary purpose of the paper is to give easy-to-use conditions for the resolution of the question.
First of all, an assumption will be described.Since in case of c = 0, every feasible solution is optimal, and since in case of c = 0 at every primal optimal solution there is at least one active index in I(x 0 ), from now on it will be assumed that I(x 0 ) = ∅.

BASIC DEFINITIONS AND THEOREMS
In nonlinear programming (NLP), in connection with optimality, the so called Kuhn-Tucker condition (KTC) plays a crucial role.It is said that the KTC is fulfilled for (P) at a feasible point x 0 iff KTC: This equation-inequality system is called the KT system.
(The corresponding NLP problem: (One can get the equivalent form by applying Motzkin's theorem of the alternative, see [2]) From now on, the phrases 'for (P)' and 'at x 0 ' will be omitted everywhere where their use is unambiguous.
A feasible point x 0 at which KTC is fulfilled is called a KT point (KTP), a vector λ satisfying KTC is called a KT vector (KTV).(Sometimes it is called a multiplier vector.)It is said that at a feasible point x 0 the strict complementary condition in connection with KTC (KT-SCC) is fulfilled iff there is a KTV with λ i > 0 for all i ∈ I(x 0 ).A feasible point x 0 at which KT-SCC is fulfilled is called a KT-SC point (KT-SCP), a vector λ satisfying the KT-SCC is called a KT-SC vector (KT-SCV).
In linear programming KTC and KTC du have the following forms.KTC: In LP the concept of strict complementarity is defined as follows.An optimal solution x 0 is said to satisfy the strict complementary condition (OPT-SCC) iff there is a dual optimal solution y 0 for which: A primal feasible point x 0 at which OPT-SCC is fulfilled is called an OPT-SC point (OPT-SCP), a dual feasible point y 0 satisfying OPT-SCC at some x 0 primal feasible point is called an OPT-SC vector (OPT-SCV).
In case of linear programming the above two kinds of strict complementarity coincide.Namely, the following assertions are true.(They are well-known and can be found in many linear programming textbooks.Therefore, their proofs are omitted from the paper.All of the following propositions pertain to linear programming.)Theorem 2.1.A primal feasible point x 0 is an optimal solution for (P) iff it is a KTP.Theorem 2.2.A dual feasible point y 0 is an optimal solution for (D) iff it is a KTV for some primal optimal solution x 0 .Theorem 2.3.For every primal optimal solution the set of KTVs is the same.
Corollary 2.4.The set of all dual optimal solutions is equal to the set of all KTVs.
Corollary 2.5.The set of all OPT-SCPs is equal to the set of all KT-SCPs.
Corollary 2.6.The set of all OPT-SCVs is equal to the set of all KT-SCVs.
Hence, OPT-SCP and KT-SCP are the same concepts, and the same is true for OPT-SCV and KT-SCV.Therefore, from now on it will be written simply SCP and SCV.
Corollary 2.7.The set of all SCPs for (P) is equal to the set of all SCVs for (D) and vice versa.
The first one of the following two theorems is called the weak complementary slackness theorem, while the second one is called the strict complementary slackness theorem.
Theorem 2.8.A primal feasible point x 0 and a dual feasible point y 0 are optimal solutions for (P) and (D), respectively iff they satisfy the condition (b − Ax 0 ) T y 0 = 0. Theorem 2.9.Let (P) or (D) has an optimal solution.Then the other has an optimal solution, too, and there are a primal optimal point x 0 and a dual optimal point y 0 with b − Ax 0 + y 0 > 0. (Hence, x 0 is a SCP and y 0 is a SCV.)

TOPOLOGICAL STRUCTURE OF THE POINTS SATISFYING STRICT COMPLEMENTARITY
In this section the topological structure of SCPs and SCVs will be investigated within the optimal solution set of (P) and (D), respectively.First, the support of a feasible solution will be defined.
The support of a primal feasible solution x 0 : supp(x 0 ) = {i ∈ I : i / ∈ I(x 0 )}, while the support of a dual feasible solution y 0 : supp(y 0 ) = {i ∈ I : i / ∈ I(y 0 )}.The next two theorems immediately follow from the strict complementary slackness theorem.Theorem 3.1.An optimal solution of (P) is a SCP iff it is an optimal point for (P) of maximal support.An optimal solution of (D) is a SCV iff it is an optimal point for (D) of maximal support.
(Maximality means that the given support is a set of maximal number of elements among the supports in question.) The following propositions show the structure of SCPs and SCVs.
Theorem 3.2.Let (P) or (D) has an optimal solution.Then there is exactly one maximal support among the supports belong to the primal optimal solutions, and there is exactly one maximal support among the supports belong to the dual optimal solutions.Proof.The theorem follows from the fact that the optimal sets of (P) and (D) are convex sets, and if e.g.x 1 and x 2 are two different primal optimal solutions of maximal support, then every inner point x 0 of the line segment between x 1 and x 2 is an optimal solution with supp(x 0 ) = supp(x 1 )∪supp(x 2 ).Corollary 3.3.SCPs are identical with the relative interior points of the primal optimal solution set (if there is any primal optimal solution) and SCVs are identical with the relative interior points of the dual optimal solution set (if there is any dual optimal solution).
Corollary 3.4.If there is only one optimal solution of (P), it is a SCP.If there is only one optimal solution of (D), it is a SCV.Corollary 3.5.Every optimal solution of (P) is a SCP iff the set of optimal solutions of (P) is nonempty and relatively open.Every optimal solution of (D) is a SCV iff the set of optimal solutions of (D) is nonempty and relatively open.

UNIQUENESS OF THE PRIMAL OPTIMAL SOLUTION
First, two conditions will be defined that are modifications of KTC.The first one is called the strict KTC (STKTC), the second one is called the weak strict KTC (WSTKTC).(Here and from now on in the paper only the equivalent dual forms of the conditions will be described.The index "du" will be omitted.)STKTC: The name WSTKTC can be explained by the dual form.It can be thought that first KTC du had been strengthened to STKTC, and then, by a weakening of STKTC, namely, substituting c T x ≤ 0 by c T x = 0 was done to get WSTKTC.)By means of these two conditions it is possible to give necessary and sufficient conditions for the uniqueness of the primal optimal solution.Theorem 4.1.A primal feasible point x 0 is the only optimal solution of (P) iff STKTC is satisfied at x 0 .
Proof. 1) Necessity.Let x 0 be the only optimal solution of (P).Then: ).On the contrary, assume that STKTC is not fulfilled at x 0 .Then there exists a vector z = 0 for which c T z ≤ 0, a T i z ≤ 0, i ∈ I(x 0 ).One can choose z in such a way that a T i z < b i − a T i x 0 could also be satisfied for all i / ∈ I(x 0 ).Then the vector x 0 + z is also an optimal solution for (P) different from x 0 , what is a contradiction to the assumption.
2) Sufficiency.Let x 0 be a primal feasible point and let STKTC be satisfied at x 0 .On the contrary, assume that it is not true that x 0 is the only optimal solution for (P).Then there exists a primal feasible solution x = x 0 for which , and this is a contradiction to the satisfaction of STKTC at x 0 .Remark 4.2.Theorem 4.1 is slightly different from the equivalence of (i) and (iii) of Theorem 2 in [3], since from Theorem 2 -writing the O matrix instead of A and −C instead of C in Theorem 2 -one can have that if x 0 is a primal optimal solution, then x 0 is the unique optimal solution of (P) iff STKTC is fulfilled.Since the preassumption in Theorem 4.1 is weaker than the one in Theorem 2, Theorem 4.1 had to be proved.Theorem 4.3.A primal optimal solution x 0 is the only optimal solution of (P) iff WSTKTC is satisfied at x 0 .
Proof.It is similar to the proof of Theorem 4.1.
The following two theorems are a little bit stronger than the equivalence of (i) and (iv), as well as that of (i) and (x) of Theorem 2 in [3].Denote by I(x 0 , y 0 ) the set I(x 0 , y 0 ) = I(x 0 ) ∩ I(y 0 ).Theorem 4.4.Let x 0 be the unique primal optimal solution.Then for every dual optimal solution y 0 : {x ∈ R n : a T i x = 0, i ∈ supp(y 0 ), a T i x ≤ 0, i ∈ I(x 0 , y 0 )} = {0}.Let x 0 be a primal optimal solution.Assume that there exists a dual optimal solution y 0 for which: {x ∈ R n : a T i x = 0, i ∈ supp(y 0 ), a T i x ≤ 0, i ∈ I(x 0 , y 0 )} = {0}.Then x 0 is the unique primal optimal solution.
Proof.It can be proved similarly as it was done in the proof in [3].
Introduce the following notation.Let A K be the sub-matrix of A consisting of those rows of A indices of which belong to the index set K, where K ⊆ I. Denote by p the cardinality of I(x 0 ), i.e. let p = card(I(x 0 )).Theorem 4.5.Let x 0 be the unique primal optimal solution.Then the columns of the matrix A I(x 0 ) are linearly independent, and for every dual optimal solution y 0 : {v ∈ R p : Let x 0 be a primal optimal solution.Assume that the columns of the matrix A I(x 0 ) are linearly independent, and there exists a dual optimal solution y 0 for which: {v ∈ R p : Proof.For the condition in Theorem 4.4: ⇔ the columns of the matrix A I(x 0 ) are linearly independent, and (The last equivalence has been got by Tucker's theorem of the alternative, see [2].)Now, it will be assumed that x 0 is a vertex of the primal feasible set.A primal feasible point x 0 is said to be a vertex of the primal feasible set iff there is no line segment belonging to the feasible set and which contains x 0 as an inner point.It is well-known that x 0 is a vertex of the primal feasible set iff x 0 is a primal feasible point and there are i 1 , i 2 , ..., i n different active indices in I(x 0 ) for which vectors a i 1 , a i 2 , ..., a in are linearly independent.(Vector a T i j is the row of A that belongs to the index i j , j = 1, 2, ..., n.)It is also widely known that if there is an optimal solution and the set of feasible solutions has a vertex, then there is an optimal solution that is a vertex.
First, three conditions, the weak KTC (WKTC), the strict weak KTC (STWKTC) and the weak strict weak KTC (WSTWKTC) will be introduced: WKTC: (The names WKTC, STWKTC and WSTWKTC can be explained in the following way.It can be thought that first KTC was weakened to the condition WKTC, then by a strengthening of WKTC, namely, substituting c T x < 0 by c T x ≤ 0 was done to get STWKTC.Finally, by weakening this last condition by substituting c T x ≤ 0 by c T x = 0, one can get WSTWKTC.)Theorem 4.6.Let x 0 be a vertex of the set of primal feasible solutions.Then it is an optimal solution of (P) iff WKTC is satisfied at x 0 .
Proof.A feasible point x 0 is an optimal solution of (P) iff it is a KTP, i.e.: x 0 is an optimal solution of (P) Since x 0 is a vertex of the primal feasible set, there are i 1 , i 2 , ..., i n different active indices in I(x 0 ) for which the vectors a i 1 , a i 2 , ..., a in are linearly independent, and hence {x ∈ R n : and hence the optimality of x 0 is equivalent to the satisfaction of WKTC.
Theorem 4.7.A primal feasible point x 0 is the only optimal solution of (P) iff it is a vertex of the primal feasible set and STWKTC is satisfied at x 0 .
Proof.By Theorem 4.1, a primal feasible point x 0 is the only optimal solution of (P 1)Sufficiency.Let x 0 be a vertex of the primal feasible set.Then -as it was proved in the proof of Theorem 4.6 -{x ∈ R n : . This is the first condition described in the beginning of the proof; and since STWKTC is just the second condition, x 0 is the only optimal solution of (P) and sufficiency has been proved.
2)Necessity.Assume that x 0 is the only optimal solution of (P).By Corollary 3.4 it is a SCP.By Corollary 3.5 the set of optimal solutions of (P) is relatively open.It could be in the only possible way that x 0 is a vertex.The satisfaction of STWKTC follows from the equivalent relations described in the beginning of the proof.Theorem 4.8.A primal optimal solution x 0 is the only optimal solution of (P) iff it is a vertex of the primal feasible set and WSTWKTC is satisfied at x 0 .Proof.By Theorem 4.3, a primal optimal solution x 0 is the only optimal solution of (P) iff {x ∈ R n : c T x = 0, a T i x ≤ 0, i ∈ I(x 0 )} = {0}.The remaining part of the proof is the same as that of the proof of Theorem 4.7, with the exception that c T x = 0 has to be written instead of c T x ≤ 0 and Theorem 4.3 has to be applied in the proof instead of Theorem 4.1.
The following theorem shows the connection between strict complementarity and WSTWKTC.Theorem 4.9.Let x 0 be a primal optimal solution.Then x 0 is a SCP iff WSTWKTC is satisfied at x 0 .
Proof.Since x 0 is a primal optimal solution, it is a KTP, too.x 0 is a SCP (by Tucker's theorem of the alternative, [2]) Since x 0 is a KTP, KTC is fulfilled.Hence, x 0 is a SCP iff WSTWKTC is satisfied at x 0 .
Corollary 4.10.A primal optimal solution x 0 is the only optimal solution of (P) iff it is a vertex of the primal feasible set and x 0 is a SCP.
Proof.It comes immediately from Theorems 4.8 and 4.9.

UNIQUENESS OF THE DUAL OPTIMAL SOLUTION
First of all, the Kyparisis' regularity condition (KYPRC) for NLP will be introduced (see [1]).KYPRC: At x 0 there is a KTV λ for which -denoting by I N N (λ(x 0 )) the set of indices of its zero components within I(x 0 ) -it is true that In nonlinear programming the following well-known assertion is true: There exists a unique KTV at x 0 iff KYPRC is fulfilled at x 0 (see [1]).The result can be applied also for LP.
Since in case of LP, by Theorem 2.1 the primal optimal solutions and KTPs are identical, and by Corollary 2.4 the dual optimal solutions and KTVs are identical, and, furthermore, by Theorem 2.3 for every primal optimal solution the set of KTVs is the same, in case of LP KYPRC has the following form, and the above assertion about the uniqueness of the KTV can be reformulated in a slightly stronger form in Theorem 5.1 as follows.KYPRC: At a given primal optimal solution x 0 there is a dual optimal solution y 0 such that or equivalently, doing the same as in the proof of Theorem 4.5, but making use of Motzkin's theorem of the alternative instead of Tucker's theorem of the alternative: KYPRC du : At a given primal optimal solution x 0 there is a dual optimal solution y 0 such that {x ∈ R n : a T i x = 0, i ∈ supp(y 0 ), a T i x > 0, i ∈ I(x 0 , y 0 )} = ∅ is satisfied and vectors a i , i ∈ supp(y 0 ) are linearly independent.
Theorem 5.1.If there are a primal optimal solution x 0 and a dual optimal solution y 0 such that KYPRC is satisfied, then y 0 is the only dual optimal solution.
If y 0 is the only dual optimal solution, then for y 0 and every primal optimal solution x 0 KYPRC is satisfied.
Corollary 5.2.If at some SCP x 0 LICQ is satisfied, then there exists a unique dual optimal solution y 0 .
If y 0 is the only dual optimal solution, then at every SCP x 0 LICQ is satisfied.
Proof.In the case when x 0 is a SCP, applying the strict complementary slackness theorem, there exists a dual optimal solution y 0 that is a SCV at x 0 .I(x 0 , y 0 ) = ∅, and hence I(x 0 ) = supp(y 0 ).Therefore, condition {x ∈ R n : a T i x = 0, i ∈ supp(y 0 ), a T i x > 0, i ∈ I(x 0 , y 0 )} = ∅ in KYPRC becomes the condition {x ∈ R n : a T i x = 0, i ∈ I(x 0 )} = ∅.But, this is always true, since 0∈ {x ∈ R n : a T i x = 0, i ∈ I(x 0 )}.Hence, in the case when x 0 is a SCP KYPRC is satisfied at x 0 iff LICQ is satisfied at x 0 .
Theorem 5.3.If there is a unique dual optimal solution, then LICQ is satisfied at every primal optimal solution.
If there is a primal optimal solution in which LICQ is satisfied, then there is a unique dual optimal solution.
Proof. 1) Assume first that there is a unique dual optimal solution.According to the duality theorem (or to Theorem 2.9) there exists a primal optimal solution.Take an arbitrary primal optimal solution x 0 .By Theorem 2.1, it is a KTP, and the KT system belonging to x 0 has a unique solution, since there is a unique dual optimal solution.From this, it follows that system { i∈I(x 0 ) λ i a i = −c} has a unique solution.Hence vectors a i , i ∈ I(x 0 ) are linearly independent.
2) Assume now that there is a primal optimal solution in which LICQ is satisfied.By Theorem 2.1, it is a KTP, and the KT system that belongs to x 0 has a solution, i.e. system { But, as LICQ is satisfied at x 0 , it has a unique solution, i.e. there is a unique KTV belonging to x 0 .Hence, by Theorems 2.3 and 2.2 there is a unique dual optimal solution.
Corollary 5.4.If there is a primal optimal solution at which LICQ is satisfied, then LICQ is satisfied at every primal optimal solution.
Proof.It follows from Theorem 5.3.
Corollary 5.5.In case of linear programming, KYPRC and LICQ are equivalent, assuming that they are investigated at a primal optimal solution.
Proof.Making use of Theorem 5.3 and Corollary 2.4 and taking into account that there is a unique KTV at x 0 iff KYPRC is satisfied at x 0 , the proof will be complete.
Theorem 5.6.Let x 0 be a primal optimal solution.If x 0 is a SCP for which SCC is satisfied with every dual optimal solution, then LICQ is fulfilled at x 0 .
Proof.According to the assumption, there is a dual optimal solution, and every dual optimal solution is a SCV.Hence, by Corollary 3.5, the set of dual optimal solutions is nonempty and relatively open.But it is well-known that an LP problem in the form of {max(−b) T y : y ∈ R m , A T y = −c, y ≥ 0} always has a feasible solution that is a vertex, provided it has a feasible solution.Therefore (see in Section 4), there is a dual optimal solution that is a vertex.Thus, the set of dual optimal solutions can not be relatively open, except for the case when it consists only of one point.In such a way, there is a unique dual optimal solution, and applying Corollary 5.2, it follows that LICQ is satisfied a x 0 .
Theorem 5.7.The conversion of Theorem 5.3 is not true, namely from the fact that at a primal optimal solution x 0 LICQ is satisfied, it does not follow that every dual optimal solution y 0 is a SCV that belongs to x 0 .
Proof.Consider the following counterexample.
Of course, x 0 = 0 ∈ R 3 is an optimal solution for (P).I(x 0 ) = {1, 2, 3} and LICQ is satisfied at x 0 .The only dual feasible point, and hence the only dual optimal solution is y T 0 = (1, 1, 0).But with x 0 and y 0 SCC is not fulfilled, since 3 ∈ I(x 0 ) and at the same time the third component of y 0 is zero.(Nevertheless, z T 0 = (0, 0, 1) is another primal optimal solution with which y 0 satisfies SCC, and LICQ is satisfied at z 0 .)

UNIQUENESS OF THE PRIMAL AND THE DUAL OPTIMAL SOLUTIONS
In the preceding two sections necessary and sufficient conditions have been shown for the uniqueness of the primal optimal solution, as well as for the dual optimal solution.In this section the two questions will be investigated simultaneously.
The first theorem uses the concepts of the vertex, SCP and LICQ.
Theorem 6.1.If there is a unique primal optimal solution x 0 and at the same time there is a unique dual optimal solution, then for x 0 the following conditions are satisfied: If there is a primal optimal solution x 0 for which the above three properties are satisfied, then x 0 is the only primal optimal solution and at the same time there is a unique dual optimal solution.Proof.Applying Corollary 4.10 and Theorem 5.3, Theorem 6.1 will be proved.
The following theorem gives a connection between LICQ and the cardinality of I(x 0 ).Theorem 6.2.Let x 0 be a vertex of the set of primal feasible solutions.Then LICQ is satisfied at x 0 iff the cardinality of I(x 0 ) is equal to n, i.e. card(I(x 0 )) = n.
Proof.Since x 0 is a vertex, there are i 1 , i 2 , ..., i n different active indices in I(x 0 ) for which vectors a i 1 , a i 2 , ..., a in are linearly independent (see Section 4).Thus {i j , j = 1, 2, . . ., n} ⊆ I(x 0 ). 1) Necessity.On the contrary, assume that card(I(x 0 )) = n.Then card(I(x 0 )) > n.But, by LICQ it would be more than n linear independent vector in R n .It is a contradiction.
In the following two propositions it is assumed that there is a primal optimal solution.Theorem 6.3.If there is a unique primal optimal solution x 0 and at the same time there is a unique dual optimal solution, then for x 0 the following conditions are satisfied: If there is a primal optimal solution x 0 for which the above three properties are satisfied, then x 0 is the only primal optimal solution and at the same time there is a unique dual optimal solution.{v ∈ R m : i∈I(x 0 ) v i a i = 0, v i ≥ 0, i ∈ I(x 0 , y 0 )} = {0} seem to be contradictory, since an apparently wider set is empty, while its subset is nonempty.However, if I(x 0 , y 0 ) = ∅, there will be no contradiction.And now that is the case.

UNIQUENESS OF THE OPTIMAL SOLUTIONS IN THE CASE OF THE SIMPLEX METHOD
In this section conditions and theorems that are pendants of the ones occurring in the preceding sections will be given for the optimal solution(s) to be unique when the primal problem is given in the form for which the primal simplex method can be applied.Thus, in this case, the LP problem is considered to be in the form: It is well-known that if there is an optimal solution for (P'), then (a proper variant of) the simplex method results an optimal solution for (P') that is a vertex of the feasible set of (P').
First, the pendants of the results got in the preceding sections for the pair of (P) and (D) will be given for the pair of (P') and (D'), and after that these new results will be specialized for the special optimal solutions got by the simplex method.(These last results are mostly widely known.) First, the pendants of some definitions occurred in the preceding sections will be described.Let x 0 be a feasible solution for (P') and y 0 for (D').For problem pair (P') and (D'): J(x 0 ) = {j ∈ J : x 0 j = 0}, J(y 0 ) = {j ∈ J : (A T y 0 ) j = c j }, where x 0 j and (A T y 0 ) j denote the jth elements of vectors x 0 and A T y 0 , respectively.LICQ: vectors a i , i ∈ I and e j , j ∈ J(x 0 ) are (together) linearly independent KTC: The definition of a KTP is the same as it was earlier.A vector (λ T , µ T ) satisfying KTC is called a KT vector (KTV).It is said that at a feasible point x 0 the strict complementary condition in connection with KTC (KT-SCC) is fulfilled iff there is a KTV with λ j > 0 for all j ∈ J(x 0 ).A feasible point x 0 at which KT-SCC is fulfilled is called a KT-SC point (KT-SCP), a vector (λ T , µ T ) satisfying the KT-SCC is called a KT-SC vector (KT-SCV).
OPT-SCC at x 0 : There is a dual optimal solution y 0 for which x T 0 (A T y 0 − c) = 0 and x 0 + A T y 0 − c > 0.
The definitions OPT-SCP and OPT-SCV are the same as earlier.
supp(x 0 ) = {j ∈ J : j / ∈ J(x 0 )} and supp(y 0 ) = {j ∈ J : j / ∈ J(y 0 )} and J(x 0 , y 0 ) = J(x 0 ) ∩ J(y 0 ).Theorem 7.1.All the propositions of Sections 2 and 3 remain valid for the case of the pair (P') and (D').The only difference is that in case of a KTV (λ T , µ T ) one has to consider its last m components as a dual optimal solution y 0 .
The definition of a vertex for (P') is the same as it was earlier for (P).It is well-known that a feasible solution for (P') is a vertex iff it is a feasible basic solution.
The pendants of the conditions introduced in Sections 4 and 5 are the following.STKTC: x j < 0, a T i x = 0, i ∈ I} = ∅ KYPRC: there is such an optimal solution y 0 of (D') that or equivalently (by Motzkin's theorem of the alternative): {x ∈ R n : a T i x = 0, i ∈ I, x j < 0, j ∈ J(x 0 , y 0 ), x j = 0, j ∈ supp(y 0 )} = ∅ and the rows of the matrix A E supp(y 0 ) are linearly independent.Having redefined all the former concepts, it is possible to give the propositions corresponding to the ones in Sections 4, 5 and 6.Denote them by writing a letter 'p' after the name of the corresponding theorem or corollary, e.g.Theorem 4.1p ('p' means pendant.)First, theorems pertaining the uniqueness of the primal optimal solution will be given.
Theorem 7.2.All the propositions of Section 4 remain valid for the case of the pair (P') and (D').The only difference is that in the case of Theorem 4.4p one must write {x ∈ R n : a T i x = 0, i ∈ I, x j ≤ 0, j ∈ J(x 0 , y 0 ), x j = 0, j ∈ supp(y 0 )} = {0} and in the case of Theorem 4.5p and the columns of matrix A belonging to the indices j ∈ J −J(x 0 ) are linearly independent' instead of the corresponding formulas in Section 4.
Proof.The proofs are similar to the ones in Section 4.Only the proofs of Theorems 4.6p, 4.7p and 4.9p have to be written down.
• Proof of Theorem 4.6p.A feasible point x 0 is an optimal solution of (P') iff it is a KTP, i.e.: x 0 is an optimal solution of (P) The last condition is WKTC.
Thus: {x ∈ R n : c T x < 0, x j = 0, j ∈ J(x 0 ), a T i x = 0, i ∈ I} = ∅, and hence x 0 is an optimal solution of (P') iff WKTC is fulfilled at x 0 .
1)Sufficiency.Similarly to the proof of Theorem 4.6p, from the fact that x 0 is a vertex, {x ∈ R n : x j = 0, j ∈ J(x 0 ), a T i x = 0, i ∈ I} = {0}.Thus: {x ∈ R n : c T x ≤ 0, x j = 0, j ∈ J(x 0 ), a T i x = 0, i ∈ I} = {0}, and hence x 0 is the only optimal solution of (P') iff STWKTC is fulfilled at x 0 .
2)The proof of necessity is the same as it is in Theorem 4.7.• Proof of Theorem 4.9p.A feasible point x 0 is an optimal solution of (P') iff it is a KTP, i.e.: x 0 is an optimal solution of (P') ⇔ {x ∈ R n : c T x < 0, x j ≤ 0, j ∈ J(x 0 ), a T i x = 0, i ∈ I} = ∅.x 0 is SCP iff {y ∈ R m : (A T y 0 ) j > c j , j ∈ J(x 0 ), (A T y 0 ) j = c j , j / ∈ J(x 0 )} = ∅.According to the nonhomogeneous generalization of Motzkin's theorem of the alternative (see [4]) the last condition is equivalent to: {x ∈ R n : c T x < 0, x j ≤ 0, j ∈ J(x 0 ), a T i x = 0, i ∈ I} = ∅ and {x ∈ R n : c T x = 0, x j ≤ 0, j ∈ J(x 0 ), j∈J(x 0 ) x j < 0, a i x = 0, i ∈ I} = ∅.
The first condition is fulfilled, since x 0 is an optimal solution of (P'), the second is WSTWKTC.Hence, in this case SCP is equivalent to WSTWKTC.
In such a way, Theorem 7.2 has been proved.
In the following theorem a proof using the propositions of the paper will be given to prove the widely known result for the uniqueness of the primal optimal solution obtained by the simplex method.
Theorem 7.3.(P') has a unique optimal solution iff the final basis B obtained by the primal simplex method as a primal optimal basis, is dual nondegenerate (i.e.all the negative reduced costs for the non-basic variables are positive).
Proof.1)Necessity.Let x 0 be the primal optimal solution obtained by the simplex method, and assume that the sufficient optimality condition of the simplex method is fulfilled in the final simplex tableau, i.e. all the negative reduced costs are non-negative.Since x 0 is a feasible basic solution for (P'), it is a vertex, and according to Corollary 4.9p, from the fact that x 0 is the only optimal solution of (P'), it follows that x 0 is a SCP.Therefore -since for all j / ∈ J B x j = 0 (where J B is the set of indices of the basic variables) -, it is necessary that for all j / ∈ J B the negative reduced cost be positive.Thus, necessity has been proved.
Consider the linear programming (LP) problem of the form (P) {min c T x : x ∈ R n , Ax ≤ b} and its dual (D) {max(−b) T y : x ∈ R m , A T y = −c, y ≥ 0}.(Here A is an m by n matrix, c ∈ R n , b ∈ R m , I = {1, 2, . . ., m} = ∅ is the set of row indices of A, J = {1, 2, . . ., n} = ∅ is the set of column indices of A.)