We are considering the application of the Augmented Lagrangian algorithms with quadratic penalty, to convex problems of quadratic programming. The problems of quadratic programming are composites of quadratic objective function and linear constraints. This important class of problems will be generated through the algorithm of sequential quadratic programming, where at each iteration the quadrat...

This paper introduces a fundamental family of unbounded convex sets that arises in the context of non-convex mixed-integer quadratic programming. It is shown that any mixed-integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a set in the family. Some fundamental properties of the convex sets are derived, along with connections to some o...

We study mixed integer nonlinear programs (MINLP) that are driven by a collection of indicator variables where each indicator variable controls a subset of the decision variables. An indicator variable, when it is “turned off”, forces some of the decision variables to assume a fixed value, and, when it is “turned on”, forces them to belong to a convex set. Most of the integer variables in known...

By using the the equivalent expression of the second order cone, we reformulate the convex second order cone programming as a boxed constrained optimization, which is a nonlinear programming with four nonnegative constraints. We give the conditions under which the stationary point of the reformulation problem solve the original problem.

The properties of the barrier F (x) = −log(det(x)), defined over the cone of squares of a Euclidean Jordan algebra, are analyzed using pure algebraic techniques. Furthermore, relating the Carathéodory number of a symmetric cone with the rank of an underlying Euclidean Jordan algebra, conclusions about the optimal parameter of F are suitably obtained. Namely, in a more direct and suitable way th...

The monotonicity of the least squares mean on the Riemannian manifold of positive definite matrices, conjectured by Bhatia and Holbrook and one of key axiomatic properties of matrix geometric means, was recently established based on the Strong Law of Large Number [14, 4]. A natural question concerned with the S.L.L.N is so called the no dice conjecture. It is a problem to make a construction of...

A function f is said to be cone superadditive if there exists a partition of R into a family of polyhedral convex cones such that f(z + x) + f(z + y) ≤ f(z) + f(z + x+ y) holds whenever x and y belong to the same cone in the family. This concept is useful in nonlinear integer programming in that, if the objective function is cone superadditive, the global minimality can be characterized by loca...

Motivated by the similarities between the properties of Z-matrices on Rn + and Lyapunov and Stein transformations on the semidefinite cone S+, we introduce and study Z-transformations on proper cones. We show that many properties of Z-matrices extend to Z-transformations. We describe the diagonal stability of such a transformation on a symmetric cone by means of quadratic representations. Final...

Interior-point methods in augmented form for linear and convex quadratic programming require the solution of a sequence of symmetric indefinite linear systems which are used to derive search directions. Safeguards are typically required in order to handle free variables or rank-deficient Jacobians. We propose a consistent framework and accompanying theoretical justification for regularizing the...