In wireless data networks the optimal routing of data depends on the link capacities which, in turn, are determined by the allocation of communications resources (such as transmit powers and bandwidths) to the links. The optimal performance of the network can only be achieved by simultaneous optimization of routing and resource allocation. In this paper, we formulate the simultaneous routing and resource allocation problem and exploit problem structure to derive ef£cient solution methods. We use a capacitated multicommodity flow model to describe the data ¤ows in the network. We assume that the capacity of a wireless link is a concave and increasing function of the communications resources allocated to the link, and the communications resources for groups of links are limited. These assumptions allow us to formulate the simultaneous routing and resource allocation problem as a convex optimization problem over the network flow variables and the communications variables. These two sets of variables are coupled only through the link capacity constraints. We exploit this separable structure by dual decomposition. The resulting solution method attains the optimal coordination of data routing in the network layer and resource allocation in the radio control layer via pricing on the link capacities.

Dantzig-Wolfe decomposition and column generation, devised for linear programs, is a success story in large scale integer programming. We outline and relate the approaches, and survey mainly recent contributions, not found in textbooks, yet. We emphasize on the growing understanding of the dual point of view, which brought considerable progress to the column generation theory and practice. It stimulated careful initializations, sophisticated solution techniques for restricted master problem and subproblem, as well as better overall performance. Thus, the dual perspective is an ever recurring concept in our "selected topics."

This paper concerns the structure of optimal codes for stochastic channel models. An investigation of an associated dual convex program reveals that the optimal distribution in channel coding is typically discrete. Based on this observation we obtain the following theoretical conclusions, as well as new algorithms for constructing capacity-achieving distributions: (i) Under general conditions, for low SNR the optimal random code is defined by a distribution whose magnitude is binary. (ii) Simple discrete approximations can nearly reach capacity even in cases where the optimal distribution is known to be absolutely continuous with respect to Lebesgue measure. (iii) A new class of algorithms is introduced, based on the cutting-plane method, to generate discrete distributions that are optimal within a prescribed class. Keywords: Information theory; channel coding; fading channels. # Department of Electrical and Computer Engineering, the Coordinated Science Laboratory, and the University of Illinois, 1308 W. Main Street, Urbana, IL 61801, URL http://black.csl.uiuc.edu:80/#meyn (s-meyn@uiuc.edu). Work supported in part by the National Science Foundation through ITR 00-85929 1

form of these problems can be described as finding a point in a nonempty bounded convex body Γ in the cone of symmetric positive semidefinite matrices. Assume that Γ is defined by an oracle, which for any given m × m symmetric positive semidefinite matrix ˆ Y either confirms that ˆ Y ∈ Γ or returns a cut, i.e., a symmetric matrix A such that Γ is in the half-space {Y: A • Y ≤ A • ˆ Y}. We study an analytic center cutting plane algorithm for this problem. At each iteration the algorithm computes an approximate analytic center of a working set defined by the cutting-plane system generated in the previous iterations. If this approximate analytic center is a solution, then the algorithm terminates; otherwise the new cutting plane returned by the oracle is added into the system. As the number of iterations increases, the working set shrinks and the algorithm eventually finds a solution of the problem. All iterates generated by the algorithm are positive definite matrices. The algorithm has a worst case complexity of O ∗ (m 3 /ɛ 2) on the total number of cuts to be used, where ɛ is the maximum radius of a ball contained by Γ.

We present an integrated procedure to build and solve big stochastic programming models. The individual components of the system --the modeling language, the solver and the hardware-- are easily accessible, or a least affordable to a large audience. The procedure is applied to a simple financial model, which can be expanded to arbitrarily large sizes by enlarging the number of scenarios. We generated a model with one million scenarios, whose deterministic equivalent linear program has 1,111,112 constraints and 2,555,556 variables. We have been able to solve it on the cluster of ten PCs in less than 3 hours. Key words. Algebraic modeling language, decomposition methods, distributed systems, large-scale optimization, stochastic programming. 1 Introduction Practical implementations of stochastic programming involve two big challenges. First, we have to build the model: its size is almost invariably large, if not huge, and this task is in itself a challenge for This research was suppor...

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Polynomial cutting plane methods based on the logarithmic barrier function and on the volumetric center are surveyed. These algorithms construct a linear programming relaxation of the feasible region, find an appropriate approximate center of the region, and call a separation oracle at this approximate center to determine whether additional constraints should be added to the relaxation. Typically, these cutting plane methods can be developed so as to exhibit polynomial convergence. The volumetric cutting plane algorithm achieves the theoretical minimum number of calls to a separation oracle. Long-step versions of the algorithms for solving convex optimization problems are presented. 1

The inclusion of transaction costs is an essential element of any realistic portfolio optimization. In this paper, we consider an extension of the standard portfolio problem in which transaction costs are incurred to rebalance an investment portfolio. The Markowitz framework of mean-variance efficiency is used with costs modelled as a percentage of the value transacted. Each security in the portfolio is represented by a pair of continuous decision variables corresponding to the amounts bought and sold. In order to properly represent the variance of the resulting portfolio, it is necessary to rescale by the funds available after paying the transaction costs. We show that the resulting fractional quadratic programming problem can be solved as a quadratic programming problem of size comparable to the model without transaction costs. Computational results for two empirical datasets are presented.

The modern era of interior-point methods dates to 1984, when Karmarkar proposed his algorithm for linear programming. In the years since then, algorithms and software for linear programming have become quite sophisticated, while extensions to more general classes of problems, such as convex quadratic programming, semidefinite programming, and nonconvex and nonlinear problems, have reached varying levels of maturity. Interior-point methodology has been used as part of the solution strategy in many other optimization contexts as well, including analytic center methods and column-generation algorithms for large linear programs. We review some core developments in the area and discuss their impact on these other problem areas.

We study the issue of updating the analytic center after multiple cutting planes have been added through the analytic center of the current polytope. This is an important issue that arises at every ‘stage ’ in a cutting plane algorithm. If q ≤ n cuts are to be added, we show that we can use a ‘Selective Orthonormalization ’ procedure to modify the cuts before adding them — it is then easy to identify a direction for an affine step into the interior of the new polytope, and the next analytic center is then found in O(q log q) Newton steps. Further, we show that multiple cut variants with selective orthonormalization of standard interior point cutting plane algorithms have the same complexity as the original algorithms.