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...

Convex regression is a promising area for bridging statistical estimation and deterministic convex optimization. New piecewise linear convex regression methods (Hannah and Dunson, 2011; Magnani and Boyd, 2009) are fast and scalable, but can have instability when used to approximate constraints or objective functions for optimization. Ensemble methods, like bagging, smearing and random partition...

In 1958, Wagner and Whitin published a seminal paper on the deterministic uncapacitated lot-sizing problem, a fundamental model that is embedded in many practical production planning problems. In this paper we consider a basic version of this model in which demands (and other problem parameters) are stochastic: the stochastic uncapacitated lot-sizing problem (SULS). We define the production pat...

A Neumann boundary control problem for a second order elliptic state equation is considered which is regularized by an energy term which is equivalent to the H−1/2(Γ)-norm of the control. Both the unconstrained and the control constrained cases are investigated. The regularity of the state, control, and co-state variables is studied with particular focus on the singularities due to the corners ...

We call a piecewise linear mapping from a planar triangulation to the plane a convex combination mapping if the image of every interior vertex is a convex combination of the images of its neighbouring vertices. Such mappings satisfy a discrete maximum principle and we show that they are oneto-one if they map the boundary of the triangulation homeomorphically to a convex polygon. This result can...

This paper addresses the problem of estimating a convex regression function under both the sup-norm risk and the pointwise risk using B-splines. The presence of the convex constraint complicates various issues in asymptotic analysis, particularly uniform convergence analysis. To overcome this difficulty, we establish the uniform Lipschitz property of optimal spline coefficients in the `∞-norm b...

This paper presents a computational approach to stability analysis of nonlinear and hybrid systems. The search for a piecewise quadratic Lyapunov function is formulated as a convex optimization problem in terms of linear matrix inequalities. The relation to frequency domain methods such as the circle and Popov criteria is explained. Several examples are included to demonstrate the flexibility a...

A class of Lyapunov functions is introduced for reaction networks satisfying simple graphical conditions. The Lyapunov functions are piecewise linear and convex in terms of the reaction rates. The existence of such functions ensures the convergence of trajectories toward the equilibria, and guarantee the asymptotic stability of the equilibria with respect to their stoichiometric compatibility c...