An analog of Karmarkar's algorithm for inequality constrained linear programs, with a "new" class of projective transformations for centering a polytope

Bibliography: p. 12.Research supported in part by ONR contract N00014-87-K-0212.by Robert M. Freund

Convergence in Karmarkar's Algorithm for Linear Programming

Karmarkar’s algorithm is formulated so as to avoid the possibility of failure because of unbounded solutions. A general inequality gives an easy proof of the convergence of the iterations. It is shown that the parameter value α = 0.5 more than doubles the originally predicted rate of convergence. To go from the last iterate to an exact optimal solution, an O(n^3) termination algorithm is prescribed. If the data have maximum bit length independent of n, the composite algorithm is shown to have complexity 0(n^4.5 log n)

Glosarium Matematika

273 p.; 24 cm

Newton's method for the general parametric center problem with applications

"March, 1991."Includes bibliographical references (p. 37-39).Kok Choon Tan and Robert M. Freund

Theoretical Efficiency of A Shifted Barrier Function Algorithm for Linear Programming

This paper examines the theoretical efficiency of solving a standard-form linear program by solving a sequence of shifted-barrier problems of the form minimize cTx - n (xj + ehj) j.,1 x s.t. Ax = b , x + e h > , for a given and fixed shift vector h > 0, and for a sequence of values of > 0 that converges to zero. The resulting sequence of solutions to the shifted barrier problems will converge to a solution to the standard form linear program. The advantage of using the shiftedbarrier approach is that a starting feasible solution is unnecessary, and there is no need for a Phase I-Phase II approach to solving the linear program, either directly or through the addition of an artificial variable. Furthermore, the algorithm can be initiated with a "warm start," i.e., an initial guess of a primal solution x that need not be feasible. The number of iterations needed to solve the linear program to a desired level of accuracy will depend on a measure of how close the initial solution x is to being feasible. The number of iterations will also depend on the judicious choice of the shift vector h . If an approximate center of the dual feasible region is known, then h can be chosen so that the guaranteed fractional decrease in e at each iteration is (1 - 1/(6 i)) , which contributes a factor of 6 ii to the number of iterations needed to solve the problem. The paper also analyzes the complexity of computing an approximate center of the dual feasible region from a "warm start," i.e., an initial (possibly infeasible) guess ir of a solution to the center problem of the dual. Key Words: linear program, interior-point algorithm, center, barrier function, shifted-barrier function, Newton step

Projective transformations for interior-point algorithms, and a superlinearly convergent algorithm for the w-center problem

Includes bibliographical references.Robert M. Freund

Polynomial-time algorithms for linear programming based only on primal scaling and projected gradients of a potential function

Bibliography: p. 28-29.Robert M. Freund

Dominating Set Games

In this paper we study cooperative cost games arising from domination problems on graphs.We introduce three games to model the cost allocation problem and we derive a necessary and su cient condition for the balancedness of all three games.Furthermore we study concavity of these games.game theory;cost allocation;cooperative games