M.L.Balinski and A.W.Tucker, "Duality theory of linear programs: A constructive approach with applications", SIAM Review 11 (1969) 347-377.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....form LP s. Our construction for the general case is closely related to the self dual form of Ye, Todd and Mizuno [15] The proofs use only analytical arguments and very little linear algebra. Our approach circumvents the use of the traditional tools such as pivoting methods (Balinski Tucker [1], Dantzig [2] Farkas Lemma (Schrijver [11] Stoer Witzgall [12] mathematical induction (Goldman Tucker [4] Tucker [14] or some general separation theorem for convex sets (von Neumann Morgenstern [9] Rockafellar [10] Farkas lemma, Motzkin s transposition theorem and Tucker s ....

Balinski, M. L. and Tucker, A. W. (1969). Duality theory of linear programs: a constructive approach with applications. SIAM Review 11, 499--581.

....well known, some of them is easily derived. The third paragraph contains the main result, a generalized criss cross method for QP and the proof of its finiteness. Finally in the fourth chapter two modifications and some special cases are presented. Concerning notations, Balinski s and Tucker s [1] notations will be used as in our previous papers [11] 12] 20] 21] 22] 23] Matrices are denoted by capital letters (A; B;P; vectors by small Latin letters and components of vectors and matrices by the corresponding Greek letters [e.g. z = i 1 ; i n ) A = ff ij ) m ....

M.L. Balinski and A.W. Tucker, "Duality theory of linear programs: A constructive approach with applications", SIAM Review 11 (1969), 347-377.

....definitions apply only to optimal faces. Another important result in linear programming is the existence of a strictly complementary optimal solution, that is, an optimal solution pair (x ; s ) such that x s 0. It is known since the early days of linear programming [16] see also [6, 53]) that such solutions exist to linear programming. It is also well known that the indices of the positive coordinates are the same for all strictly complementary pairs. We denote by B f1; 2; ng the indices of positive coordinates of x . Similarly, N denotes the indices of coordinates ....

.... optimal face, and hence provide an optimal solution with maximal number of nonzero coordinates to both the primal and dual problem (see e.g. Guler and Ye [24] The existence of strictly complementary primal dual optimal solutions has been proved first by Goldman and Tucker [16] Balinski Tucker [6] give a (non polynomial) algorithm to generate this such a strictly complementary pair. In contrast, as we will see below, an optimal basis can be obtained from an optimal primal dual solution pair in strongly polynomial time. Since IPM s provide a maximal complementary optimal solution pair, a ....

Balinski, M. L. and Tucker, A. W. (1969). Duality theory of linear programs: A constructive approach with applications. SIAM Review 11(3), 499--581.

....(see e.g. Guler and Ye [34] It is also proved that these solutions are strictly complementary and they define the optimal partition of the LP problem. The existence of strictly complementary primal dual optimal solutions has been proved first by Goldman and Tucker [27] Balinski and Tucker [9] propose a pivot algorithm to generate such a strictly complementary pair. In contrast, as we will see later on, an optimal basis can be obtained from any primal dual optimal solution pair in strongly polynomial time. To have an optimal basis might be important for several reasons. For example, ....

Balinski, M. L. and Tucker, A. W. (1969) Duality theory of linear programs: A constructive approach with applications. SIAM Review 11, 499--581.

....be defined later) This algorithm is a generalization of the algorithm of Megiddo. Although our algorithm is of both theoretical and practical interest, we will mainly be concerned with the theoretical aspects. For LP there also exists a TI algorithm. This algorithm is due to Balinski and Tucker [3]. Balinski and Tucker used this algorithm or procedure to proof the existence of strictly complementary solutions for LP by constructing so called Balinski Tucker tableaus. In this paper their approach is generalized to QP and LCPs with sufficient matrices to obtain a TI algorithm. To our best ....

....algorithm to solve such problems exists up till date. However, if we have any complementary solution, the algorithm can be used to find a complementary basis in strongly polynomial time. 5 Tripartition Identification (TI) Algorithm 5. 1 Balinski Tucker tableaus for LCPs Balinski and Tucker [3] introduced a staircase shaped optimal tableau for linear programming. This tableau is called a Balinski Tucker tableau. Balinski and Tucker showed that it is possible to find a strictly complementary solution of an LP from such a Balinski Tucker tableau very easily by performing some additions ....

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Balinski, M.L. and Tucker, A.W. (1969), Duality Theory of Linear Programs: A Constructive Approach with Applications, SIAM Review 11, 347--377.

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M.L.Balinski and A.W.Tucker, "Duality theory of linear programs: A constructive approach with applications", SIAM Review 11 (1969) 347-377.

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