Semidefinite programming (SDP) is a subfield of convex optimization concerned with the optimization of a linear objective function (a user-specified function that the user wants to minimize or maximize)
over the intersection of the cone of positive semidefinite matrices with an affine space, i.e., a spectrahedron.
Semidefinite programming is a relatively new field of optimization which is of growing interest for several reasons. Many practical problems in operations research and combinatorial optimization can be modeled or approximated as semidefinite programming problems. In automatic control theory, SDPs are used in the context of linear matrix inequalities. SDPs are in fact a special case of cone programming and can be efficiently solved by interior point methods.
All linear programs and (convex) quadratic programs can be expressed as SDPs, and via hierarchies of SDPs the solutions of polynomial optimization problems can be approximated. Semidefinite programming has been used in the optimization of complex systems. In recent years, some quantum query complexity problems have been formulated in terms of semidefinite programs.
A linear programming problem is one in which we wish to maximize or minimize a linear objective function of real variables over a polytope. In semidefinite programming, we instead use real-valued vectors and are allowed to take the dot product of vectors; nonnegativity constraints on real variables in LP (linear programming) are replaced by semidefiniteness constraints on matrix variables in SDP (semidefinite programming). Specifically, a general semidefinite programming problem can be defined as any mathematical programming problem of the form
where the , and the are real numbers and is the dot product of and .
An matrix is said to be positive semidefinite if it is the Gram matrix of some vectors (i.e. if there exist vectors such that for all ). If this is the case, we denote this as . Note that there are several other equivalent definitions of being positive semidefinite, for example, positive semidefinite matrices are self-adjoint matrices that have only non-negative eigenvalues.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
In computational complexity theory, the unique games conjecture (often referred to as UGC) is a conjecture made by Subhash Khot in 2002. The conjecture postulates that the problem of determining the approximate value of a certain type of game, known as a unique game, has NP-hard computational complexity. It has broad applications in the theory of hardness of approximation. If the unique games conjecture is true and P ≠ NP, then for many important problems it is not only impossible to get an exact solution in polynomial time (as postulated by the P versus NP problem), but also impossible to get a good polynomial-time approximation.
In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. If the primal is a minimization problem then the dual is a maximization problem (and vice versa). Any feasible solution to the primal (minimization) problem is at least as large as any feasible solution to the dual (maximization) problem.
Semidefinite programming (SDP) is a subfield of convex optimization concerned with the optimization of a linear objective function (a user-specified function that the user wants to minimize or maximize) over the intersection of the cone of positive semidefinite matrices with an affine space, i.e., a spectrahedron. Semidefinite programming is a relatively new field of optimization which is of growing interest for several reasons. Many practical problems in operations research and combinatorial optimization can be modeled or approximated as semidefinite programming problems.
The students gain an in-depth knowledge of several current and emerging areas of theoretical computer science. The course familiarizes them with advanced techniques, and develops an understanding of f
This course introduces the theory and applications of optimization. We develop tools and concepts of optimization and decision analysis that enable managers in manufacturing, service operations, marke
Covers solving integer linear programs graphically, algorithmically, and through optimization methods.
,
A new approach is presented to obtain a convex set of robust D—stabilizing fixed structure controllers, relying on Cauchy's argument principle. A convex set of D—stabilizing controllers around an init
We describe a series of algorithms that efficiently implement Gaussian model-X knockoffs to control the false discovery rate on large-scale feature selection problems. Identifying the knockoff distrib
SIAM PUBLICATIONS2021
, ,
We propose a stochastic conditional gradient method (CGM) for minimizing convex finitesum objectives formed as a sum of smooth and non-smooth terms. Existing CGM variants for this template either suff