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17
Symmetric quasidefinite matrices
 SIAM Journal on Optimization
, 1995
"... We say that a symmetric matrix K is quasidefinite if it has the form ..."
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Cited by 52 (3 self)
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We say that a symmetric matrix K is quasidefinite if it has the form
Computational Experience with Approximation Algorithms for the Set Covering Problem
, 1994
"... The Set Covering problem (SCP) is a well known combinatorial optimization problem, which is NPhard. We conducted a comparative study of eight different approximation algorithms for the SCP, including several greedy variants, fractional relaxations, randomized algorithms and a neural network algorit ..."
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Cited by 42 (2 self)
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The Set Covering problem (SCP) is a well known combinatorial optimization problem, which is NPhard. We conducted a comparative study of eight different approximation algorithms for the SCP, including several greedy variants, fractional relaxations, randomized algorithms and a neural network algorithm. The algorithms were tested on a set of randomgenerated problems with up to 500 rows and 5000 columns, and on two sets of problems originating in combinatorial questions with up to 28160 rows and 11264 columns. On the random problems and on one set of combinatorial problems, the best algorithm among those we tested was the neural network algorithm, with greedy variants very close in second and third place. On the other set of combinatorial problems, the best algorithm was a greedy variant and the neural network performed quite poorly. The other algorithms we tested were always inferior to the ones mentioned above. Theoretical Division and CNLS, MS B213 Los Alamos National Lab, Los Ala...
Lowauthority controller design via convex optimization
 AIAA Journal of Guidance, Control, and Dynamics
, 1999
"... In this paper we address the problem of lowauthority 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 closedloop eigenvalues can be well approximated analytically using perturbati ..."
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Cited by 31 (14 self)
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In this paper we address the problem of lowauthority 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 closedloop eigenvalues can be well approximated analytically using perturbation theory. These analytical approximations may suffice to predict the behavior of the closedloop 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 interiorpoint 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: Lowauthority control, actuator/sensor placement, linear operator perturbation theory, convex optimization, secondorder cone programming, semidefinite programming, linear matrix inequality. 1
Practical Problem Solving with Cutting Plane Algorithms in Combinatorial Optimization
, 1994
"... Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely comb ..."
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Cited by 20 (5 self)
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Cutting plane algorithms have turned out to be practically successful computational tools in combinatorial optimization, in particular, when they are embedded in a branch and bound framework. Implementations of such "branch and cut" algorithms are rather complicated in comparison to many purely combinatorial algorithms. The purpose of this article is to give an introduction to cutting plane algorithms from an implementor's point of view. Special emphasis is given to control and data structures used in practically successful implementations of branch and cut algorithms. We also address the issue of parallelization. Finally, we point out that in important applications branch and cut algorithms are not only able to produce optimal solutions but also approximations to the optimum with certified good quality in moderate computation times. We close with an overview of successful practical applications in the literature.
Stability Of Linear Equations Solvers In InteriorPoint Methods
 SIAM J. Matrix Anal. Appl
, 1994
"... . Primaldual interiorpoint methods for linear complementarity and linear programming problems solve a linear system of equations to obtain a modified Newton step at each iteration. These linear systems become increasingly illconditioned in the later stages of the algorithm, but the computed steps ..."
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Cited by 16 (2 self)
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. Primaldual interiorpoint methods for linear complementarity and linear programming problems solve a linear system of equations to obtain a modified Newton step at each iteration. These linear systems become increasingly illconditioned in the later stages of the algorithm, but the computed steps are often sufficiently accurate to be useful. We use error analysis techniques tailored to the special structure of these linear systems to explain this observation and examine how theoretically superlinear convergence of a pathfollowing algorithm is affected by the roundoff errors. Key words. primaldual interiorpoint methods, error analysis, stability AMS(MOS) subject classifications. 65G05, 65F05, 90C33 1. Introduction. The monotone linear complementarity problem (LCP) is the problem of finding a vector pair (x; y) 2 R l n \Theta R l n such that y = Mx+ q; (x; y) 0; x T y = 0; (1) where M (a real, n \Theta n positive semidefinite matrix) and q (a real vector with n elements...
Stability of Augmented System Factorizations in InteriorPoint Methods
 SIAM J. Matrix Anal. Appl
, 1997
"... . Some implementations of interiorpoint algorithms obtain their search directions by solving symmetric indefinite systems of linear equations. The conditioning of the coefficient matrices in these socalled augmentedsystems deteriorates on later iterations, as some of the diagonal elements grow wit ..."
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Cited by 14 (2 self)
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. Some implementations of interiorpoint algorithms obtain their search directions by solving symmetric indefinite systems of linear equations. The conditioning of the coefficient matrices in these socalled augmentedsystems deteriorates on later iterations, as some of the diagonal elements grow without bound. Despite this apparent difficulty, the steps produced by standard factorization procedures are often accurate enough to allow the interiorpoint method to converge to high accuracy. When the underlying linear program is nondegenerate, we show that convergence to arbitrarily high accuracy occurs, at a rate that closely approximates the theory. We also explain and demonstrate what happens when the linear program is degenerate, where convergence to acceptable accuracy (but not arbitrarily high accuracy) is usually obtained. 1. Introduction. We focus on the core linear algebra operation in primaldual interiorpoint methods for linear programming: solution of a system of linear equat...
A heuristic for optimizing stochastic activity networks with applications to statistical digital circuit sizing
 IEEE Transactions on Circuits and SystemsI
, 2004
"... A deterministic activity network (DAN) is a collection of activities, each with some duration, along with a set of precedence constraints, which specify that activities begin only when certain others have finished. One critical performance measure for an activity network is its makespan, which is th ..."
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Cited by 12 (4 self)
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A deterministic activity network (DAN) is a collection of activities, each with some duration, along with a set of precedence constraints, which specify that activities begin only when certain others have finished. One critical performance measure for an activity network is its makespan, which is the minimum time required to complete all activities. In a stochastic activity network (SAN), the durations of the activities and the makespan are random variables. The analysis of SANs is quite involved, but can be carried out numerically by Monte Carlo analysis. This paper concerns the optimization of a SAN, i.e., the choice of some design variables that affect the probability distributions of the activity durations. We concentrate on the problem of minimizing a quantile (e.g., 95%) of the makespan, subject to constraints on the variables. This problem has many applications, ranging from project management to digital integrated circuit (IC) sizing (the latter being our motivation). While there are effective methods for optimizing DANs, the SAN optimization problem is much more difficult; the few existing methods cannot handle largescale problems.
Current Trends in Stochastic Programming Computation and Applications
, 1995
"... While decisions frequently have uncertain consequences, optimal decision models often replace those uncertainties with averages or best estimates. Limited computational capability may have motivated this practice in the past. Recent computational advances have, however, greatly expanded the range of ..."
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Cited by 10 (0 self)
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While decisions frequently have uncertain consequences, optimal decision models often replace those uncertainties with averages or best estimates. Limited computational capability may have motivated this practice in the past. Recent computational advances have, however, greatly expanded the range of stochastic programs, optimal decision models with explicit consideration of uncertainties. This paper describes basic methodology in stochastic programming, recent developments in computation, and some practical application examples.
Solving Problems with Semidefinite and Related Constraints Using InteriorPoint Methods for Nonlinear Programming. this volume
, 2002
"... Abstract. In this paper, we describe how to reformulate a problem that has secondorder cone and/or semidefiniteness constraints in order to solve it using a generalpurpose interiorpoint algorithm for nonlinear programming. The resulting problems are smooth and convex, and numerical results from t ..."
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Cited by 8 (1 self)
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Abstract. In this paper, we describe how to reformulate a problem that has secondorder cone and/or semidefiniteness constraints in order to solve it using a generalpurpose interiorpoint algorithm for nonlinear programming. The resulting problems are smooth and convex, and numerical results from the DIMACS Implementation Challenge problems and SDPLib are provided. 1.
Control applications of nonlinear convex programming
 the 1997 IFAC Conference on Advanced Process Control
, 1998
"... Since 1984 there has been a concentrated e ort to develop e cient interiorpoint methods for linear programming (LP). In the last few years researchers have begun to appreciate a very important property of these interiorpoint methods (beyond their e ciency for LP): they extend gracefully to nonline ..."
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Cited by 6 (3 self)
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Since 1984 there has been a concentrated e ort to develop e cient interiorpoint methods for linear programming (LP). In the last few years researchers have begun to appreciate a very important property of these interiorpoint methods (beyond their e ciency for LP): they extend gracefully to nonlinear convex optimization problems. New interiorpoint algorithms for problem classes such as semide nite programming (SDP) or secondorder cone programming (SOCP) are now approaching the extreme e ciency of modern linear programming codes. In this paper we discuss three examples of areas of control where our ability to e ciently solve nonlinear convex optimization problems opens up new applications. In the rst example we show how SOCP can be used to solve robust openloop optimal control problems. In the second example, we show how SOCP can be used to simultaneously design the setpoint and feedback gains for a controller, and compare this method with the more standard approach. Our nal application concerns analysis and synthesis via linear matrix inequalities and SDP. Submitted to a special issue of Journal of Process Control, edited by Y. Arkun & S. Shah, for papers presented at the 1997 IFAC Conference onAdvanced Process Control, June 1997, Ban. This and related papers available via anonymous FTP at