this paper can be downloaded from http://www.compgeom.com/meb/. P. Kumar and J. Mitchell are partially supported by a grant from the National Science Foundation (CCR0098172) . J. Mitchell is also partially supported by grants from the Honda Fundamental Research Labs, Metron Aviation, NASAAmes Research (NAG2-1325), and the US-Israel Binational Science Foundation. E. A. Yldrm is partially supported by an NSF CAREER award (DMI-0237415)

this paper. 1.9.1 The spectral bundle method

Conventional methods for optimal sizing of wires and transistors use linear RC circuit models and the Elmore delay as a measure of signal delay. If the RC circuit has a tree topology the sizing problem reduces to a convex optimization problem which can be solved using geometric programming. The tree topology restriction precludes the use of these methods in several sizing problems of significant importance to high-performance deep submicron design including, for example, circuits with loops of resistors, e.g., clock distribution meshes, and circuits with coupling capacitors, e.g., buses with crosstalk between the lines. The paper proposes a new optimization method which can be used to address these problems. The method uses the dominant time constant as a measure of signal propagation delay in an RC circuit, instead of Elmore delay. Using this measure, sizing of any RC circuit can be cast as a convex optimization problem which can be solved using the recently developed efficient interi...

Abstract—This paper describes a linear matrix inequality (LMI)-based algorithm for the static and reduced-order output-feedback synthesis problems of nth-order linear time-invariant (LTI) systems with nu (respectively, ny) independent inputs (respectively, outputs). The algorithm is based on a “cone complementarity ” formulation of the problem and is guaranteed to produce a stabilizing controller of order m n 0 max(nu;ny), matching a generic stabilizability result of Davison and Chatterjee [7]. Extensive numerical experiments indicate that the algorithm finds a controller with order less than or equal to that predicted by Kimura’s generic stabilizability result (m n0nu0ny+1). A similar algorithm can be applied to a variety of control problems, including robust control synthesis. Index Terms — Complementarity problem, linear matrix inequality, reduced-order stabilization, static output feedback. I.

. Many problems in engineering analysis and design can be cast as convex optimization problems, often nonlinear and nondifferentiable. We give a high-level description of recently developed interior-point methods for convex optimization, explain how problem structure can be exploited in these algorithms, and illustrate the general scheme with numerical experiments. To give a rough idea of the efficiencies obtained, we are able to solve convex optimization problems with over 1000 variables and 10000 constraints in around 10 minutes on a workstation. Keywords. Optimization, numerical methods, linear programming, optimal control, robust control, convex programming, interior-point methods, FIR filter design, conjugate gradients 1. INTRODUCTION Many problems in engineering analysis and design can be cast as convex optimization problems, i.e., min f 0 (x) s.t. f i (x) 0; i = 1; : : : ; L; where the functions f i are convex. It is widely known that such problems have desirable properties,...

In this paper we address the problem of low-authority controller (LAC) design. The premise is that the actuators have limited authority, and hence cannot significantly shift the eigenvalues of the system. As a result, the closed-loop eigenvalues can be well approximated analytically using perturbation theory. These analytical approximations may suffice to predict the behavior of the closed-loop system in practical cases, and will provide at least a very strong rationale for the first step in the design iteration loop. We will show that LAC design can be cast as convex optimization problems that can be solved efficiently in practice using interior-point methods. Also, we will show that by optimizing the ℓ1 norm of the feedback gains, we can arrive at sparse designs, i.e., designs in which only a small number of the control gains are nonzero. Thus, in effect, we can also solve actuator/sensor placement or controller architecture design problems. Keywords: Low-authority control, actuator/sensor placement, linear operator perturbation theory, convex optimization, second-order cone programming, semi-definite programming, linear matrix inequality. 1

This paper considers the problem of minimizing a linear function over the intersection of an affine space with a closed convex cone. In the first half of the paper, we give a detailed study of duality properties of this problem and present examples to illustrate these properties. In particular, we introduce the notions of weak/strong feasibility or infeasibility for a general primal-dual pair of conic convex programs, and then establish various relations between these notions and the duality properties of the problem. In the second half of the paper, we propose a self-dual embedding with the following properties: Any weakly centered sequence converging to a complementary pair either induces a sequence converging to a certificate of strong infeasibility, or induces a sequence of primaldual pairs for which the amount of constraint violation converges to zero, and the corresponding objective values are in the limit not worse than the optimal objective value(s). In case of strong duality, ...

this paper, we are only concerned about the last question, which can be answered using semidefinite programming. For a survey of other aspects of such geometric representations, see [64].

Aircraft conflict detection and resolution is currently attracting the interest of many air transportation service providers and is concerned with the following question: Given a set of airborne aircraft and their intended trajectories, what control strategy should be followed by the pilots and the air traffic service provider to prevent the aircraft from coming too close to each other? This paper addresses this problem by presenting a distributed air-ground architecture, whereby each aircraft proposes its desired heading while a centralized air traffic control architecture resolves any conflict arising between the aircraft involved in the conflict, while minimizing the deviation between desired and conflict-free heading for each aircraft. The resolution architecture relies on a combination of convex programming and randomized searches: It is shown that aversion of the planar, multi-aircraft conflict resolution problem that accounts for all possible crossing patterns among aircraft might be recast as a nonconvex, quadratically constrained quadratic program. For this type of problem, there exist efficient numerical relaxations, based on semidefinite programming, that provide lower bounds

In Kernel Fisher discriminant analysis (KFDA), we carry out Fisher linear discriminant analysis in a high dimensional feature space defined implicitly by a kernel. The performance of KFDA depends on the choice of the kernel; in this paper, we consider the problem of finding the optimal kernel, over a given convex set of kernels. We show that this optimal kernel selection problem can be reformulated as a tractable convex optimization problem which interior-point methods can solve globally and efficiently. The kernel selection method is demonstrated with some UCI machine learning benchmark examples. 1.