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CSDP, a C library for semidefinite programming.
, 1997
"... this paper is organized as follows. First, we discuss the formulation of the semidefinite programming problem used by CSDP. We then describe the predictor corrector algorithm used by CSDP to solve the SDP. We discuss the storage requirements of the algorithm as well as its computational complexity. ..."
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Cited by 206 (2 self)
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this paper is organized as follows. First, we discuss the formulation of the semidefinite programming problem used by CSDP. We then describe the predictor corrector algorithm used by CSDP to solve the SDP. We discuss the storage requirements of the algorithm as well as its computational complexity. Finally, we present results from the solution of a number of test problems. 2 The SDP Problem We consider semidefinite programming problems of the form max tr (CX)
Moment Problems and Semidefinite Optimization
 WORKING PAPER, SLOAN SCHOOL OF MANAGEMENT, MIT
, 2000
"... Problems involving moments of random variables arise naturally in many areas of mathematics, economics, and operations research. How dowe obtain optimal bounds on the probability that a random variable belongs in a set, given some of its moments? How dowe price financial derivatives without assuming ..."
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Cited by 12 (0 self)
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Problems involving moments of random variables arise naturally in many areas of mathematics, economics, and operations research. How dowe obtain optimal bounds on the probability that a random variable belongs in a set, given some of its moments? How dowe price financial derivatives without assuming any model for the underlying price dynamics, given only moments of the price of the underlying asset? How do we obtain stronger relaxations for stochastic optimization problems exploiting the knowledge that the decision variables are moments of random variables? Can we generate near optimal solutions for a discrete optimization problem from a semidefinite relaxation by interpreting an optimal solution of the relaxation as a covariance matrix? In this paper, we demonstrate that convex, and in particular semidefinite, optimization methods lead to interesting and often unexpected answers to these questions.
Control applications of nonlinear convex programming
 SUBMITTED TO A SPECIAL ISSUE OF JOURNAL OF PROCESS CONTROL, EDITED BY Y. ARKUN & S. SHAH
, 1998
"... Since 1984 there has been a concentrated effort to develop efficient 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 efficiency for LP): they extend gracefully to no ..."
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Cited by 8 (3 self)
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Since 1984 there has been a concentrated effort to develop efficient 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 efficiency for LP): they extend gracefully to nonlinear convex optimization problems. New interiorpoint algorithms for problem classes such as semidefinite programming (SDP) or secondorder cone programming (SOCP) are now approaching the extreme efficiency of modern linear programming codes. In this paper we discuss three examples of areas of control where our ability to efficiently solve nonlinear convex optimization problems opens up new applications. In the first 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 final application concerns analysis and synthesis via linear matrix inequalities and SDP.
Parallel Implementation of Successive Convex Relaxation Methods for . . .
 J. OF GLOBAL OPTIMIZATION
, 2002
"... As computing resources continue to improve, global solutions for larger size quadraticallyconstrained optimization problems become more achievable. In this paper, we focus on larger size problems and get accurate bounds for optimal values of such problems with the successive use of SDP relaxations ..."
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Cited by 2 (2 self)
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As computing resources continue to improve, global solutions for larger size quadraticallyconstrained optimization problems become more achievable. In this paper, we focus on larger size problems and get accurate bounds for optimal values of such problems with the successive use of SDP relaxations on a parallel computing system called Ninf (Network based Information Library for high performance computing).
LowAuthority Controller Design by Means of Convex Optimization
, 1999
"... this paper we introduce a new method for lowauthority controller design, based on convex programming. We formulate the LAC design problem as a nonlinear convex optimization problem, which can then be solved ef# ciently by interiorpoint methods. The advantage of formulating the problem as convex is ..."
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this paper we introduce a new method for lowauthority controller design, based on convex programming. We formulate the LAC design problem as a nonlinear convex optimization problem, which can then be solved ef# ciently by interiorpoint methods. The advantage of formulating the problem as convex is that largeorder problems can be solved (globally) in practice. Another advantage of this formulation is that it can handle a wide variety of speci# cations and objectives beyond standard eigenvalue placement. Typical design objectives for the LAC design include increased damping or decay rate for the system response, and typical constraints include limitations on the controller gains and actuator power. We show that, by optimizing the l 1 norm of the 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. Moreover, it is possible to address the robustness of the LAC, i.e., closedloop performance subject to uncertainties or variations in the plant model. Therefore, by combining all these, for example, we can solve the problem of robust actuator/sensor placement and LAC design in one step