The main purpose of this paper is to extend the conventional separation theorems concerning the convex subset of Rm to generalized separation theorems concerning the convex subset of Rm × Sn. This is accomplished by the introduction of the generalized inner product in Rm × Sn. Then we derive the famous Farkas' lemma in nonlinear semidefinite programming, which may be very important for the anal...

In this paper we consider minimizing the spectral condition number of a positive semidefinite matrix over a nonempty closed convex set Ω. We show that it can be solved as a convex programming problem, and moreover, the optimal value of the latter problem is achievable. As a consequence, when Ω is positive semidefinite representable, it can be cast into a semidefinite programming problem. We the...

Many combinatorial optimization problems have relaxations that are semidefinite programming problems. In principle, the combinatorial optimization problem can then be solved by using a branch-and-cut procedure, where the problems to be solved at the nodes of the tree are semidefinite programs. It is desirable that the solution to one node of the tree should be exploited at the child node in ord...

A significant special case of the problems which could be solved were those whose constraints were given by semidefinite cones. A Semidefinite Program (SDP) is an optimisation over the intersection of an affine set and cone of positive semidefinite matrices (Alizadeh and Goldfarb, 2001). Cone programming is discussed more in Section 3. Within semidefinite programming there is a smaller set of p...

A central drawback of primal-dual interior point methods for semidefinite programs is their lack of ability to exploit problem structure in cost and coefficient matrices. This restricts applicability to problems of small dimension. Typically semidefinite relaxations arising in combinatorial applications have sparse and well structured cost and coefficient matrices of huge order. We present a me...

An important class of optimization problems in control and signal processing involves the constraint that a Popov function is nonnegative on the unit circle or the imaginary axis. Such a constraint is convex in the coefficients of the Popov function. It can be converted to a finitedimensional linear matrix inequality via the Kalman-Yakubovich-Popov lemma. However, the linear matrix inequality r...

In a common formulation of semi-infinite programs, the infinite constraint set is a requirement that a function parametrized by the decision variables is nonnegative over an interval. If this function is sufficiently closely approximable by a polynomial or a rational function, then the semi-infinite program can be reformulated as an equivalent semidefinite program. Solving this semidefinite pro...

Spectrahedra are sets defined by linear matrix inequalities. Projections of spectrahedra are called semidefinite representable sets. Both kinds of sets are of practical use in polynomial optimization, since they occur as feasible sets in semidefinite programming. There are several recent results on the question which sets are semidefinite representable. So far, all results focus on the case of ...

In this paper, we present a novel semidefinite programming approach for multiple-instance learning. We first formulate the multipleinstance learning as a combinatorial maximummargin optimization problem with additional instance selection constraints within the framework of support vector machines. Although solving this primal problem requires non-convex programming, we nevertheless can then der...

An important class of optimization problems in control and signal processing involves the constraint that a Popov function is nonnegative on the unit circle or the imaginary axis. Such a constraint is convex in the coefficients of the Popov function. It can be converted to a finitedimensional linear matrix inequality via the Kalman-Yakubovich-Popov lemma. However, the linear matrix inequality r...