We consider contraction of convex hypersurfaces by convex speeds, homogeneous of degree one in the principal curvatures, that are not necessarily smooth. We show how to approximate such a speed by a sequence of smooth speeds for which behaviour is well known. By obtaining speed and curvature pinching estimates for the flows by the approximating speeds, independent of the smoothing parameter, we...

In this paper, we aim to prove the linear rate convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex composite optimization problems. Under a mild calmness condition, which holds automatically for convex composite piecewise linear-quadratic programming, we establish the global Q-linear rate of convergence for a general semi-proximal ADMM w...

A branch and bound global optimization method, BB, for general continuous optimization problems involving nonconvexities in the objective function and/or constraints is presented. The nonconvexities are categorized as being either of special structure or generic. A convex relaxation of the original nonconvex problem is obtained by (i) replacing all nonconvex terms of special structure (i.e. bil...

Many crucial results of the asymptotic theory of symmetric convex bodies were extended to the non-symmetric case in recent years. That led to the conjecture that for every n-dimensional convex body K there exists a projection P of rank k, proportional to n, such that PK is almost symmetric. We prove that the conjecture does not hold. More precisely, we construct an n-dimensional convex body K s...

The graduated optimization approach, also known as the continuation method, is a popular heuristic to solving non-convex problems that has received renewed interest over the last decade. Despite being popular, very little is known in terms of its theoretical convergence analysis. In this paper we describe a new first-order algorithm based on graduated optimization and analyze its performance. W...

At the apex of an idiopathic scoliotic curve there is a greater proportion of "slow twitch" muscle fibres in multifidus on the convex as compared to the concave side. To determine whether this represents a primary muscular imbalance relevant to the aetiology of idiopathic scoliosis or merely a secondary change, the lengths of multifidus on opposite sides of the curve were measured. Multifidus i...

This paper presents the formulation of a flight clearance criterion as a convex optimization problem. The criterion is the stability margins criterion which is specified as an allowable phase and gain margin of a certain loop transfer function. The satisfaction of the criterion amounts to the Nichols plot of the loop transfer function being outside a specified trapezoidal region. It was shown p...

This paper deals with the problem of local exponential stochastic stabilization of continuous time Bilinear Active Fault Tolerant Control Systems with Markovian Parameters (BAFTCSMP). The design technique is based on a differential inclusion of the bilinear term for a restricted domain of the state space. The above problematic is addressed under a convex programming approach. Indeed, conditions...

The decomposition method is currently one of the major methods for solving the convex quadratic optimization problems being associated with support vector machines. For a special case of such problems the convergence of the decomposition method to an optimal solution has been proven based on a working set selection via the gradient of the objective function. In this paper we will show that a ge...

This paper looks at axioms for convexity, and shows how they can be applied to discrete spaces. Two structures for a discrete geometry are considered: oriented matroids, and cell complexes. Oriented matroids are shown to have a structure which naturally satisfies the axioms for being a convex geometry. Cell complexes are shown to give rise to various different notions of convexity, one of which...