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Author Index Volume 83
https://ddd.uab.cat/record/174
Mon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1741998Solving stochastic programs with integer recourse by enumeration: : A framework using Gröbner basis reductions
https://ddd.uab.cat/record/173
In this paper we present a framework for solving stochastic programs with complete integer recourse and discretely distributed right-hand side vector, using Gröbner basis methods from computational algebra to solve the numerous second-stage integer programs. Using structural properties of the expected integer recourse function, we prove that under mild conditions an optimal solution is contained in a finite set. Furthermore, we present a basic scheme to enumerate this set and suggest improvements to reduce the number of function evaluations needed. . Schultz, RüdigerMon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1731998Infeasible-interior-point paths for sufficient linear complementarity problems and their analyticity
https://ddd.uab.cat/record/172
In this paper we study the behavior of infeasible-interior-point-paths for solving horizontal linear complementarity problems that are sufficient in the sense of Cottle et al. (R. W. Cottle, J. -S. Pang, Venkateswaran, Linear Algebra Appl. 114/115 (1989) 231-249). We show that these paths converge to a central point of the set of solutions. It is also shown that these are analytic functions of the path parameter even at the limitpoint, if the complementarity problem has a strictly complementary solution, and have a simple branchpoint, if it is solveable, but has no strictly complementarity solution. . Stoer, JosefMon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1721998Discrete convex analysis
https://ddd.uab.cat/record/171
A theory of "discrete convex analysis" is developed for integer-valued functions defined on integer lattice points. The theory parallels the ordinary convex analysis, covering discrete analogues of the fundamental concepts such as conjugacy, subgradients, the Fenchel min-max duality, separation theorems and the Lagrange duality framework for convex/nonconvex optimization. The technical development is based on matroid-theoretic concepts, in particular, submodular functions and exchange axioms. Sections 1-4 extend the conjugacy relationship between submodularity and exchange ability, deepening our understanding of the relationship between convexity and submodularity investigated in the eighties by A. Frank, S. Fujishige, L. Lovász and others. Sections 5 and 6 establish duality theorems for M- and L-convex functions, namely, the Fenchel min-max duality and separation theorems. These are the generalizations of the discrete separation theorem for submodular functions due to A. Frank and the optimality criteria for the submodular flow problem due to M. Iri-N. Tomizawa, S. Fujishige, and A. Frank. A novel Lagrange duality framework is also developed in integer programming. We follow Rockafellar's conjugate duality approach to convex/nonconvex programs in nonlinear optimization, while technically relying on the fundamental theorems of matroid-theoretic nature. . Murota, KazuoMon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1711998A branch and bound method for stochastic global optimization
https://ddd.uab.cat/record/170
A stochastic branch and bound method for solving stochastic global optimization problems is proposed. As in the deterministic case, the feasible set is partitioned into compact subsets. To guide the partitioning process the method uses stochastic upper and lower estimates of the optimal value of the objective function in each subset. Convergence of the method is proved and random accuracy estimates derived. Methods for constructing stochastic upper and lower bounds are discussed. The theoretical considerations are illustrated with an example of a facility location problem. . Norkin, Vladimir I.Mon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1701998Affine scaling algorithm fails for semidefinite programming
https://ddd.uab.cat/record/169
In this paper, we introduce an affine scaling algorithm for semidefine programming (SDP), and give an example of a semidefinite program such that the affine scaling algorithm converges to a non-optimal point. Both our program and its dual have interior feasible solutions and unique optimal solutions which satisfy strict complementarity, and they are non-degenerate everywhere. . Muramatsu, MasakazuMon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1691998L-shaped decomposition of two-stage stochastic programs with integer recourse
https://ddd.uab.cat/record/168
We consider two-stage stochastic programming problems with integer recourse. The L-shaped method of stochastic linear programming is generalized to these problems by using generalized Benders decomposition. Nonlinear feasibility and optimality cuts are determined via general duality theory and can be generated when the second stage problem is solved by standard techniques. Finite convergence of the method is established when Gomory's fractional cutting plane algorithm or a branch-and-bound algorithm is applied. . Carøe, Claus C.Mon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1681998Location and shape of a rectangular facility in R^n. Convexity properties
https://ddd.uab.cat/record/167
In this paper we address a generalization of the Weber problem, in which we seek for the center and the shape of a rectangle (the facility) minimizing the average distance to a given set (the demand-set) which is not assumed to be finite. Some theoretical properties of the average distance are studied, and an expression for its gradient, involving solely expected distances to rectangles, is obtained. This enables the resolution of the problem by standard optimization techniques. . Carrizosa, E.Mon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1671998Perturbation analysis of a condition number for convex inequality systems and global error bounds for analytic systems
https://ddd.uab.cat/record/166
In this paper several types of perturbations on a convex inequality system are considered, and conditions are obtained for the system to be well-conditioned under these types of perturbations, where the well-conditionedness of a convex inequality system is defined in terms of the uniform boundedness of condition numbers under a set of perturbations. It is shown that certain types of perturbations can be used to characterize the well-conditionedness of a convex inequality system, in which either the system has a bounded solution set and satisfies the Slater condition or an associated convex inequality system, which defines the recession cone of the solution set for the system, satisfies the Slater condition. Finally, sufficient conditions are given for the existence of a global error bound for an analytic system. It is shown that such a global error bound always holds for any inequality system defined by finitely many convex analytic functions when the zero vector is in the relative interior of the domain of an associated convex conjugate function. . Sien, DengMon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1661998Merit functions for semi-definite complementarity problems
https://ddd.uab.cat/record/165
Merit functions such as the gap function, the regularized gap function, the implicit Lagrangian, and the norm squared of the Fischer-Burmeister function have played an important role in the solution of complementarity problems defined over the cone of nonnegative real vectors. We study the extension of these merit functions to complementarity problems defined over the cone of block-diagonal symmetric positive semi-definite real matrices. The extension suggests new solution methods for the latter problems. . Tseng, PaulMon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1651998Error bounds for nondifferentiable convex inequalities under a strong Slater constraint qualification
https://ddd.uab.cat/record/164
A global error bound is given on the distance between an arbitrary point in the n-dimensional real space R^n and its projection on a nonempty convex set determined by m convex, possibly nondifferentiable, inequalities. The bound is in terms of a natural residual that measures the violations of the inequalities multiplied by a new simple condition constant that embodies a single strong Slater constraint qualification (CQ) which implies the ordinary Slater CQ. A very simple bound on the distance to the projection relative to the distance to a point satisfying the ordinary Slater CQ is given first and then used to derive the principal global error bound. . Mangasarian, O.L.Mon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1641998Author Index Volume 76
https://ddd.uab.cat/record/163
Mon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1631997Author Index Volume 77
https://ddd.uab.cat/record/162
Mon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1621997Predictor-corrector algorithm for solving P_* (k)-matrix LCP from arbitrary positive starting points
https://ddd.uab.cat/record/161
A new predictor-corrector algorithm is proposed for solving P_*(k)-matrix linear complementarity problems. If the problem is solvable, then the algorithm converges from an arbitrary positive starting point (x^0, s^0). The computational complexity of the algorithm depends on the quality of the starting point. If the starting point is feasible or close to being feasible, it has O((1 + k) V~n/(rho)_0L)-iteration complexity, where (rho)_0 is the ratio of the smallest and average coordinate of X^0s^0. With appropriate initialization, a modified version of the algorithm terminates in O((1 + k)^2 (n/(rho)_0)L) steps either by finding a solution or by determining that the problem has no solution in a predetermined, arbitrarily large, region. The algorithm is quadratically convergent for problems having a strictly complementary solution. We also propose an extension of a recent algorithm of Mizuno to P_* (k)-matrix linear complementarity problems such that it can start from arbitrary positive points and has superlinear convergence without a strictly complementary condition. . Potra, Florian A.Mon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1611997Author Index Volume 79
https://ddd.uab.cat/record/160
Mon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1601997Author Index Volume 78
https://ddd.uab.cat/record/159
Mon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1591997Nominations for 1997 Elections
https://ddd.uab.cat/record/158
Dennis, J.Mon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1581997Duality theorem for a generalized Fermat-Weber problem
https://ddd.uab.cat/record/157
The classical Fermat-Weber problem is to minimize the sum of the distances from a point in a plane to k given points in the plane. This problem was generalized by Witzgall to n-dimensional space and to allow for a general norm, not necessarily symmetric; he found a dual for this problem. The authors generalize this result further by proving a duality theorem which includes as special cases a great variety of choices of norms in the terms of the Fermat-Weber sum. The theorem is proved by applying a general duality theorem of Rockafellar. As applications, a dual is found for the multi-facility location problem and a nonlinear dual is obtained for a linear programming problem with a priori bounds for the variables. When the norms concerned are continuously differentiable, formulas are obtained for retrieving the solution for each primal problem from the solution of its dual. . Kaplan, WilfredMon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1571997Plant location with minimum inventory
https://ddd.uab.cat/record/156
We present an integer programming model for plant location with inventory costs. The linear programming relaxation has been solved by Dantzig-Wolfe decomposition. In this case the subproblems reduce to the minimum cut problem. We have used subgradient optimization to accelerate the convergence of the D-W algorithm. We present our experience with problems arising in the design of a distribution network for computer spare parts. In most cases, from a fractional solution we were able to derive integer solutions within 4% of optimality. . Barahona, FranciscoMon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1561998Worst-case analyses, linear programming and the bin-packing problem
https://ddd.uab.cat/record/155
In this paper we consider the familiar bin-packing problem and its associated set-partitioning formulation. We show that the optimal solution to the bin-packing problem can be no larger than 4/3 Z _LP where Z_LP is the optimal solution value of the linear programming relaxation of the set-partitioning formulation. An example is provided to show that the bound is tight. A by-product of our analysis is a new worst-case bound on the performance of the well studied First Fit Decreasing and Best Fit Decreasing heuristics. Chan, Lap Mui AnnMon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1551998Condition measures and properties of the central trajectory of a linear program
https://ddd.uab.cat/record/154
Given a data instance d = (A, b, c) of a linear program, we show that certain properties of solutions along the central trajectory of the linear program are inherently related to the condition number C(d) of the data instance d = (A, b, c), where C(d) is a scale-invariant reciprocal of a closely-related measure (rho)(d) called the "distance to ill-posedness". (The distance to ill-posedness essentially measures how close the data instance d = (A, b, c) is to being primal or dual infeasible. ) We present lower and upper bounds on sizes of optimal solutions along the central trajectory, and on rates of change of solutions along the central trajectory, as either the barrier parameter µ or the data d = (A, b, c) of the linear program is changed. These bounds are all linear or polynomial functions of certain natural parameters associated with the linear program, namely the condition number C(d), the distance to ill-posedness (rho)(d), the norm of the data d and the dimensions m and n. Nunez, Manuel A.Mon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1541998A bundle-Newton method for nonsmooth unconstrained minimization
https://ddd.uab.cat/record/153
An algorithm based on a combination of the polyhedral and quadratic approximation is given for finding stationary points for unconstrained minimization problems with locally Lipschitz problem functions that are not necessarily convex or differentiable. Global convergence of the algorithm is established. Under additional assumptions, it is shown that the algorithm generates Newton iterations and that the convergence is superlinear. Some encouraging numerical experience is reported. . Luksan, LadislavMon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1531998On proving existence of feasible points in equality constrained optimization problems
https://ddd.uab.cat/record/152
Various algorithms can compute approximate feasible points or approximate solutions to equality and bound constrained optimization problems. In exhaustive search algorithms for global optimizers and other contexts, it is of interest to construct bounds around such approximate feasible points, then to verify (computationally but rigorously) that an actual feasible point exists within these bounds. Hansen and others have proposed techniques for proving the existence of feasible points within given bounds, but practical implementations have not, to our knowledge, previously been described. Various alternatives are possible in such an implementation, and details must be carefully considered. Also, in addition to Hansen's technique for handling the underdetermined case, it is important to handle the overdetermined case, when the approximate feasible point corresponds to a point with many active bound constraints. The basic ideas, along with experimental results from an actual implementation, are summarized here. . Kearfott, R. BakerMon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1521998Approximate iterations in Bregman-function-based proximal algorithms
https://ddd.uab.cat/record/151
This paper establishes convergence of generalized Bregman-function-based proximal point algorithms when the iterates are computed only approximately. The problem being solved is modeled as a general maximal monotone operator, and need not reduce to minimization of a function. The accuracy conditions on the iterates resemble those required for the classical "linear" proximal point algorithm, but are slightly stronger; they should be easier to verify or enforce in practice than conditions given in earlier analyses of approximate generalized proximal methods. Subjects to these practically enforceable accuracy restrictions, convergence is obtained under the same conditions currently established for exact Bregman-function-based proximal methods. . Eckstein, JonathanMon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1511998A variable-penalty alternating directions method for convex optimization
https://ddd.uab.cat/record/150
We study a generalized version of the method of alternating directions as applied to the minimization of the sum of two convex functions subject to linear constraints. The method consists of solving consecutively in each iteration two optimization problems which contain in the objective function both Lagrangian and proximal terms. The minimizers determine the new proximal terms and a simple update of the Lagrangian terms follows. We prove a convergence theorem which extends existing results by relaxing the assumption of uniqueness of minimizers. Another novelty is that we allow penalty matrices, and these may vary per iteration. This can be beneficial in applications, since it allows additional tuning of the method to the problem and can lead to faster convergence relative to fixed penalties. As an application, we derive a decomposition scheme for block angular optimization and present computational results on a class of dual block angular problems. . Kontogiorgis, SpyridonMon, 13 Mar 2006 16:31:26 GMThttps://ddd.uab.cat/record/1501998