Lieven Vandenberghe

Learn More
In sernidefinite programming, one minimizes a linear function subject to the constraint that an affine combination ofsynunetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. Semidefinite programming unifies several standard problems (e.g., linear and(More)
A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form. Recently developed solution methods can solve even large-scale GPs extremely efficiently and reliably; at the same time a number of practical problems, particularly in circuit design, have been found to be(More)
The problem of maximizing the determinant of a matrix subject to linear matrix inequalities arises in many elds, including computational geometry, statistics, system identi cation, experiment design, and information and communication theory. It can also be considered as a generalization of the semide nite programming problem. We give an overview of the(More)
Semide nite programming (or SDP) has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, control theory, and statistics. This tremendous research(More)
The nuclear norm (sum of singular values) of a matrix is often used in convex heuristics for rank minimization problems in control, signal processing, and statistics. Such heuristics can be viewed as extensions of l1-norm minimization techniques for cardinality minimization and sparse signal estimation. In this paper we consider the problem of minimizing(More)
We discuss convex optimization problems where some of the variables are constrained to be finite autocorrelation sequences. Problems of this form arise in signal processing and communications, and we describe applications in filter design and system identification. Autocorrelation constraints in optimization problems are often approximated by sampling the(More)
We consider the design of nite impulse response (FIR) lters subject to upper and lower bounds on the frequency response magnitude. The associated optimization problems, with the lter coe cients as the variables and the frequency response bounds as constraints, are in general nonconvex. Using a change of variables and spectral factorization, we can pose such(More)
We present a new semide nite programming approach to FIR lter design with arbitrary upper and lower bounds on the frequency response magnitude. It is shown that the constraints can be expressed as linear matrix inequalities (LMIs), and hence they can be easily handled by recent interior-point methods. Using this LMI formulation, we can cast several(More)