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12
On a Homogeneous Algorithm for the Monotone Complementarity Problem
 Mathematical Programming
, 1995
"... We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility and compleme ..."
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Cited by 24 (3 self)
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We present a generalization of a homogeneous selfdual linear programming (LP) algorithm to solving the monotone complementarity problem (MCP). The algorithm does not need to use any "bigM" parameter or twophase method, and it generates either a solution converging towards feasibility and complementarity simultaneously or a certificate proving infeasibility. Moreover, if the MCP is polynomially solvable with an interior feasible starting point, then it can be polynomially solved without using or knowing such information at all. To our knowledge, this is the first interiorpoint and infeasiblestarting algorithm for solving the MCP that possesses these desired features. Preliminary computational results are presented. Key words: Monotone complementarity problem, homogeneous and selfdual, infeasiblestarting algorithm. Running head: A homogeneous algorithm for MCP. Department of Management, Odense University, Campusvej 55, DK5230 Odense M, Denmark, email: eda@busieco.ou.dk. y De...
On HomotopySmoothing Methods for Variational Inequalities
"... A variational inequality problem with a mapping g : ! n ! ! n and lower and upper bounds on variables can be reformulated as a system of nonsmooth equations F (x) = 0 in ! n . Recently, several homotopy methods, such as interiorpoint and smoothing methods, have been employed to solve the prob ..."
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Cited by 23 (5 self)
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A variational inequality problem with a mapping g : ! n ! ! n and lower and upper bounds on variables can be reformulated as a system of nonsmooth equations F (x) = 0 in ! n . Recently, several homotopy methods, such as interiorpoint and smoothing methods, have been employed to solve the problem. All of these methods use parametric functions and construct perturbed equations to approximate the problem. The solution to the perturbed system constitutes a smooth trajectory leading to the solution of the original variational inequality problem. The methods generate iterates to follow the trajectory. Among these methods ChenMangasarian and GabrielMor'e proposed a class of smooth functions to approximate F . In this paper, we study several properties of the trajectory defined by solutions of these smooth systems. We propose a homotopysmoothing method for solving the variational inequality problem, and show that the method converges globally and superlinearly under mild conditions. ...
An Interior Point Potential Reduction Method for Constrained Equations
, 1995
"... We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In gen ..."
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Cited by 11 (3 self)
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We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In general, constrained equations provide a unified formulation for many mathematical programming problems, including complementarity problems of various kinds and the KarushKuhnTucker systems of variational inequalities and nonlinear programs. Combining ideas from the damped Newton and interior point methods, we present an iterative algorithm for solving a constrained system of equations and investigate its convergence properties. Specialization of the algorithm and its convergence analysis to complementarity problems of various kinds and the KarushKuhnTucker systems of variational inequalities are discussed in detail. We also report the computational results of the implementation of the algo...
Polynomiality of PrimalDual Affine Scaling Algorithms for Nonlinear Complementarity Problems
, 1995
"... This paper provides an analysis of the polynomiality of primaldual interior point algorithms for nonlinear complementarity problems using a wide neighborhood. A condition for the smoothness of the mapping is used, which is related to Zhu's scaled Lipschitz condition, but is also applicable to mappi ..."
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Cited by 10 (4 self)
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This paper provides an analysis of the polynomiality of primaldual interior point algorithms for nonlinear complementarity problems using a wide neighborhood. A condition for the smoothness of the mapping is used, which is related to Zhu's scaled Lipschitz condition, but is also applicable to mappings that are not monotone. We show that a family of primaldual affine scaling algorithms generates an approximate solution (given a precision ffl) of the nonlinear complementarity problem in a finite number of iterations whose order is a polynomial of n, ln(1=ffl) and a condition number. If the mapping is linear then the results in this paper coincide with the ones in [13].
Linear Algebra for Semidefinite Programming
, 1995
"... Let M n (IK) denote the set of all n 2 n matrices with elements in IK, where IK represents the field IR of real numbers, the field 0 C of complex numbers or the (noncommutative) field IH of quaternion numbers. We call a subset T of M n (IK) a *subalgebra of M n (IK) over the field IR (or simply a ..."
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Cited by 8 (4 self)
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Let M n (IK) denote the set of all n 2 n matrices with elements in IK, where IK represents the field IR of real numbers, the field 0 C of complex numbers or the (noncommutative) field IH of quaternion numbers. We call a subset T of M n (IK) a *subalgebra of M n (IK) over the field IR (or simply a *subalgebra) if (i) T forms a subring of M n (IK) with the usual addition A + B and multiplication AB of matrices A; B 2 M n (IK); specifically the zero matrix O and the identity matrix I belong to T . (ii) T is an IRmodule, i.e., a vector space over the field IR; ffA + fiB 2 T for every ff; fi 2 IR and A; B 2 T , (iii) A 3 2 T if A 2 T , where A 3 denotes the conjugate transpose of A 2 M n (IK). The introduction of *subalgebras T provides us with a unified and compact way of handling LPs (linear programs) in IR n , SDPs (semidefinite programs) in M n (IR), M n ( 0 C) and M n (IH), and monotone SDLCPs (semidefinite linear complementarity problems) in those spaces. We can extend t...
On Superlinear Convergence of InfeasibleInteriorPoint Algorithms for Linearly Constrained Convex Programs
 Computational Optimization and Applications
, 1996
"... This note derives bounds on the length of the primaldual affine scaling directions associated with a linearly constrained convex program satisfying the following conditions: 1) the problem has a solution satisfying strict complementarity, 2) the Hessian of the objective function satisfies a certain ..."
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Cited by 2 (0 self)
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This note derives bounds on the length of the primaldual affine scaling directions associated with a linearly constrained convex program satisfying the following conditions: 1) the problem has a solution satisfying strict complementarity, 2) the Hessian of the objective function satisfies a certain invariance property. We illustrate the usefulness of these bounds by establishing the superlinear convergence of the algorithm presented in Wright and Ralph [22] for solving the optimality conditions associated with a linearly constrained convex program satisfying the above conditions. 1 Introduction During the past few years, we have seen the appearance of many papers dealing with primaldual (feasible and infeasible) interior point algorithms for linear programs (LP), convex quadratic programs (QP), monotone linear complementarity problems (LCP) and monotone nonlinear complementarity problems (NCP) that are superlinearly or quadratically convergent. For LP and QP, these works include [1,...
Global Linear And Local Quadratic Convergence Of A LongStep AdaptiveMode Interior Point Method For Some Monotone Variational Inequality Problems
, 1996
"... . An interior point method is proposed to solve variational inequality problems for monotone functions and polyhedral sets. The method has the following advantages. 1. Given an initial interior feasible solution with duality gap ¯ 0 , the algorithm requires at most O[n log(¯ 0 =ffl)] iterations to ..."
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Cited by 2 (0 self)
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. An interior point method is proposed to solve variational inequality problems for monotone functions and polyhedral sets. The method has the following advantages. 1. Given an initial interior feasible solution with duality gap ¯ 0 , the algorithm requires at most O[n log(¯ 0 =ffl)] iterations to obtain an ffloptimal solution. 2. The rate of convergence of the duality gap is qquadratic. 3. At each iteration, a longstep improvement based on a line search is allowed. 4. The algorithm can automatically transfer from a linear mode to a quadratic mode to accelerate the local convergence. Keywords: Polynomial Complexity of Algorithms, Interior Point Methods, Monotone Variational Inequality Problems, Rate of Convergence. 1 The research is partially supported by Grant RP930033 of National University of Singapore. 2 Department of Decision Sciences. Email: fbasunj@nus.sg. 3 Department of Mathematics. Email: matzgy@nus.sg. 1 Introduction Given a function F : IR n ! IR n and a nonem...
A Quadratically Convergent Polynomial LongStep Algorithm For A Class Of Nonlinear Monotone Complementarity Problems
, 1999
"... . Several interior point algorithms have been proposed for solving nonlinear monotone complementarity problems. Some of them have polynomial worstcase complexity but have to confine to short steps, whereas some of the others can take long steps but no polynomial complexity is proven. This paper pre ..."
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Cited by 1 (0 self)
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. Several interior point algorithms have been proposed for solving nonlinear monotone complementarity problems. Some of them have polynomial worstcase complexity but have to confine to short steps, whereas some of the others can take long steps but no polynomial complexity is proven. This paper presents an algorithm which is both longstep and polynomial. In addition, the sequence generated by the algorithm, as well as the corresponding complementarity gap, converges quadratically. The proof of the polynomial complexity requires that the monotone mapping satisfies a scaled Lipschitz condition, while the quadratic rate of convergence is derived under the assumptions that the problem has a strictly complementary solution and that the Jacobian of the mapping satisfies certain regularity conditions. Keywords: Complexity of Algorithms, Interior Point Methods, Monotone Complementarity Problems, Rate of Convergence. 1 The research is partially supported by Grant RP930033 of National Universi...
Lexicographic Maxmin Fairness in a Wireless Adhoc Network with Random
 Access,In Proceedings of IEEE Conference on Decision and Control (CDC),San Diego
, 2006
"... We consider the lexicographic maxmin fair rate control problem at the link layer in a random access wireless network. In lexicographic maxmin fair rate allocation, the minimum link rates are maximized in a lexicographic order. For the Aloha multiple access model, we propose iterative approaches th ..."
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Cited by 1 (0 self)
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We consider the lexicographic maxmin fair rate control problem at the link layer in a random access wireless network. In lexicographic maxmin fair rate allocation, the minimum link rates are maximized in a lexicographic order. For the Aloha multiple access model, we propose iterative approaches that attain the optimal rates under very general assumptions on the network topology and communication pattern; the approaches are also amenable to distributed implementation. The algorithms and results in this paper generalize those in our previous work [7] on maximizing the minimum rates in a random access network, and nicely connects to the “bottleneckbased ” lexicographic rate optimization algorithm popularly used in wired networks [1]. 1
Summary of Research (19942010)
"... The results of the research over the past 17 years are contained in 81 papers published in professional journals, and in 20 papers published in refereed proceedings. In what follows we summarize some of the main results. Modeling atmospheric chemistry and air pollution My interest in this area start ..."
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The results of the research over the past 17 years are contained in 81 papers published in professional journals, and in 20 papers published in refereed proceedings. In what follows we summarize some of the main results. Modeling atmospheric chemistry and air pollution My interest in this area started at the “Center for Global and Regional Environmental Research” (CGRER) of the University of Iowa. There I worked in an interdisciplinary team together with Gregory Carmichael from the Chemical Engineering Department and several Ph.D. students in chemical engineering, computer science, and applied mathematics. The goal of that group was to develop a “Third Generation Model ” for atmospheric chemistry. The requirements for such a model had been formulated by a panel of scientists (including Gregory Carmichael and myself) assembled by EPA [54]. It involved setting up a parabolic system consisting of about 300 nonlinear PDEs and finding efficient numerical methods for solving it. This is a 3D reactive transport problem with about 100 chemical species and 200 chemical reactions. Since transport and chemical reactions operate at different time scales, an operator split technique is applied with different integration methods for transport and chemistry. In fact chemistry is solved by integrating at each point of the space grid a (very stiff) initial value ODE problem for a period of time equal to the time step of the transport integrator. Since up to 90 % of computer time in atmospheric chemistry simulations is used in integrating chemistry, our group initially focussed its efforts on improving the chemistry integrators. This research effort lead to the reduction of the chemistry integration time by an order of magnitude and contributed to a better theoretical understanding of the behavior of numerical integrators for chemical rate equations.