A basic algorithm for the minimization of a differentiable convex function (in particular, a strictly convex quadratic function) defined on the convex hull of m points in R is outlined. Each iteration of the algorithm is implemented in barycentric coordinates, the number of which is equal tom. The method is based on a new procedure for finding the projection of the gradient of the objective fun...

In this paper, a new model based on differential-algebraic equations(DAEs) for solving convex quadratic programming(CQP) problems is proposed. It is proved that the new approach is guaranteed to generate optimal solutions for this class of optimization problems. This paper also shows that the conventional interior point methods for solving (CQP) problems can be viewed as a special case of the n...

Goal: Describe the image of the cone Sn ≥� of positive semidefinite quadratic forms under the projection πG . Sn ≥�: convex cone of quadratic forms∑i , j ai jxix j such that∑i , j ai jpi p j ≥ � for all (p�, . . . , pn) ∈ Rn. Theorem (Diagonalization ofQuadratic Forms). Aquadratic form q ∈ R[x�, . . . , xn] is positive semidefinite if and only if it is a sum of squares of linear forms after a c...

The problem of minimizing a quadratic form over the unit simplex, referred to as a standard quadratic optimization problem, admits an exact reformulation as a linear optimization problem over the convex cone of completely positive matrices. This computationally intractable cone can be approximated from the inside and from the outside by two sequences of nested polyhedral cones of increasing acc...

We consider a square-integrable semimartingale and investigate the convex order relations between its discrete, continuous and predictable quadratic variation. As the main results, we show that if the semimartingale has conditionally independent increments and symmetric jump measure, then its discrete realized variance dominates its quadratic variation in increasing convex order. The results ha...

In this paper, under a suitable regularity condition, we establish that a broad class of conic convex polynomial optimization problems, called conic sum-of-squares convex polynomial programs, exhibits exact conic programming relaxation, which can be solved by various numerical methods such as interior point methods. By considering a general convex cone-program, we give unified results that appl...

Estimating a vector from noisy quadratic observations is a task that arises naturally in many contexts, from dimensionality reduction, to synchronization and phase retrieval problems. It is often the case that additional information is available about the unknown vector (for instance, sparsity, sign or magnitude of its entries). Many authors propose non-convex quadratic optimization problems th...

We propose a quartic function to represent a family of continuous quality loss functions. Depending on the choice of its parameters the shape of this function within the specification limits can be either symmetric or asymmetric, and it can be either similar to the ubiquitous quadratic loss function or somewhat closer to the conventional step function. We examine this family of loss functions i...

Scalarization of the multiobjective programming problems with fuzzy coefficients using the embedding theorem and the concept of convex cone (ordering cone) is proposed in this paper. Since the set of all fuzzy numbers can be embedded into a normed space, this motivation naturally inspires us to invoke the scalarization techniques in vector optimization problems to evaluate the multiobjective pr...

Let (QP ) be a 0-1 quadratic program which consists in minimizing a quadratic function subject to linear constraints. In this paper, we present a general method to solve (QP ) by reformulation of the problem into an equivalent 0-1 program with a convex quadratic objective function, followed by the use of a standard mixed integer quadratic programming solver. Our convexification method, which is...