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219
Convex analysis on the Hermitian matrices
 SIAM Journal on Optimization
, 1996
"... There is growing interest in optimization problems with real symmetric matrices as variables. Generally the matrix functions involved are spectral: they depend only on the eigenvalues of the matrix. It is known that convex spectral functions can be characterized exactly as symmetric convex functions ..."
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Cited by 42 (17 self)
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There is growing interest in optimization problems with real symmetric matrices as variables. Generally the matrix functions involved are spectral: they depend only on the eigenvalues of the matrix. It is known that convex spectral functions can be characterized exactly as symmetric convex functions of the eigenvalues. A new approach to this characterization is given, via a simple Fenchel conjugacy formula. We then apply this formula to derive expressions for subdifferentials, and to study duality relationships for convex optimization problems with positive semidefinite matrices as variables. Analogous results hold for Hermitian matrices. Key Words: convexity, matrix function, Schur convexity, Fenchel duality, subdifferential, unitarily invariant, spectral function, positive semidefinite programming, quasiNewton update. AMS 1991 Subject Classification: Primary 15A45 49N15 Secondary 90C25 65K10 1 Introduction A matrix norm on the n \Theta n complex matrices is called unitarily inv...
Power control by geometric programming
 IEEE Trans. on Wireless Commun
, 2005
"... Abstract — In wireless cellular or ad hoc networks where Quality of Service (QoS) is interferencelimited, a variety of power control problems can be formulated as nonlinear optimization with a systemwide objective, e.g., maximizing the total system throughput or the worst user throughput, subject ..."
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Cited by 40 (5 self)
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Abstract — In wireless cellular or ad hoc networks where Quality of Service (QoS) is interferencelimited, a variety of power control problems can be formulated as nonlinear optimization with a systemwide objective, e.g., maximizing the total system throughput or the worst user throughput, subject to QoS constraints from individual users, e.g., on data rate, delay, and outage probability. We show that in the high SignaltoInterference Ratios (SIR) regime, these nonlinear and apparently difficult, nonconvex optimization problems can be transformed into convex optimization problems in the form of geometric programming; hence they can be very efficiently solved for global optimality even with a large number of users. In the medium to low SIR regime, some of these constrained nonlinear optimization of power control cannot be turned into tractable convex formulations, but a heuristic can be used to compute in most cases the optimal solution by solving a series of geometric programs through the approach of successive convex approximation. While efficient and robust algorithms have been extensively studied for centralized solutions of geometric programs, distributed algorithms have not been explored before. We present a systematic method of distributed algorithms for power control that is geometricprogrammingbased. These techniques for power control, together with their implications to admission control and pricing in wireless networks, are illustrated through several numerical examples. Index Terms — Convex optimization, CDMA power control, Distributed algorithms. I.
Antenna Array Pattern Synthesis via Convex Optimization
, 1997
"... 'We show that a variety of antenna array pattern synthesis problems can be expressed as convex optimization problems, which can be (numerically) solved with great efficiency by recently developed interiorpoint methods. The synthesis problems involve arrays with arbitrary geometry and element direct ..."
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Cited by 37 (8 self)
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'We show that a variety of antenna array pattern synthesis problems can be expressed as convex optimization problems, which can be (numerically) solved with great efficiency by recently developed interiorpoint methods. The synthesis problems involve arrays with arbitrary geometry and element directivity, constraints on far and nearfield patterns over narrow or broad frequency bandwidth, and some important robustness constraints. We show several numerical simulations for the particular problem of constraining the beampattern level of a simple array for adaptive and broadband arrays.
A cone Complementarity Linearization Algorithm for Static OutputFeedback and Related Problems
 IEEE Transaction on Automatic Control
, 1997
"... Abstract—This paper describes a linear matrix inequality (LMI)based algorithm for the static and reducedorder outputfeedback synthesis problems of nthorder linear timeinvariant (LTI) systems with nu (respectively, ny) independent inputs (respectively, outputs). The algorithm is based on a “cone ..."
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Cited by 35 (0 self)
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Abstract—This paper describes a linear matrix inequality (LMI)based algorithm for the static and reducedorder outputfeedback synthesis problems of nthorder linear timeinvariant (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, reducedorder stabilization, static output feedback. I.
Approximate Minimum Enclosing Balls in High Dimensions Using CoreSets
, 2003
"... 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 Resear ..."
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Cited by 34 (8 self)
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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 (NAG21325), and the USIsrael Binational Science Foundation. E. A. Yldrm is partially supported by an NSF CAREER award (DMI0237415)
Bundle Methods to Minimize the Maximum Eigenvalue Function
, 1999
"... this paper. 1.9.1 The spectral bundle method ..."
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
An introduction to convex optimization for communications and signal processing
 IEEE J. Sel. Areas Commun
, 2006
"... Abstract—Convex optimization methods are widely used in the ..."
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Cited by 29 (2 self)
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Abstract—Convex optimization methods are widely used in the
GPCAD: A Tool for CMOS OpAmp Synthesis
 IN PROCEEDINGS OF THE IEEE/ACM INTERNATIONAL CONFERENCE ON COMPUTER AIDED DESIGN
, 1998
"... We present a method for optimizing and automating component and transistor sizing for CMOS operational amplifiers. We observe that a wide variety of performance measures can be formulated as posynomial functions of the design variables. As a result, amplifier design problems can be formulated as a g ..."
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Cited by 29 (10 self)
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We present a method for optimizing and automating component and transistor sizing for CMOS operational amplifiers. We observe that a wide variety of performance measures can be formulated as posynomial functions of the design variables. As a result, amplifier design problems can be formulated as a geometric program, a special type of convex optimization problem for which very efficient global optimization methods have recently been developed. The synthesis method is therefore fast, and determines the globally optimal design; in particular the final solution is completely independent of the starting point (which can even be infeasible), and infeasible specifications are unambiguously detected. After briefly
Semidefinite programs and combinatorial optimization (Lecture notes)
, 1995
"... 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]. ..."
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Cited by 29 (1 self)
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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].