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In sernidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of synunetric 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)

- Miguel Sousa Lobo, Lieven Vandenberghe, Stephen Boyd, Herv E Lebret
- 1998

In a second-order cone program (SOCP) a linear function is minimized over the intersection of an aane set and the product of second-order (quadratic) cones. SOCPs are nonlinear convex problems that include linear and (convex) quadratic programs as special cases, but are less general than semideenite programs (SDPs). Several eecient primal-dual… (More)

This is a collection of additional exercises, meant to supplement those found in the book Convex Optimization, by Stephen Boyd and Lieven Vandenberghe. These exercises were used in several courses on convex optimization, EE364a (Stanford), EE236b (UCLA), or 6.975 (MIT), usually for homework, but sometimes as exam questions. Some of the exercises were… (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)

- Erna Unrau, ROMESH SAIGAL, LIEVEN VANDENBERGHE, Henry Wolkowicz, Romesh Saigal, Lieven Vandenberghe +1 other

The problem of maximizing the determinant of a matrix subject to linear matrix inequalities arises in many elds, including computational geometry, statistics, system identiication, experiment design, and information and communication theory. It can also be considered as a generalization of the semideenite programming problem. We give a n o v erview of the… (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 ℓ 1-norm minimization techniques for cardinality minimization and sparse signal estimation. In this paper we consider the problem of minimizing… (More)

A sharp lower bound on the probability of a set defined by quadratic inequalities, given the first two moments of the distribution, can be efficiently computed using convex optimization. This result generalizes Chebyshev's inequality for univariate random variables. Two semidefinite programming formulations are presented, with a constructive proof based on… (More)

We present a new semideenite 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)