We propose a sequential method to estimate monotone convex function that consists of: (i) monotone regression via solving a constrained least square problem and (ii) convexification of the monotone regression estimate via solving an associated constrained uniform approximation problem. We show that this method is faster than the constrained least squares (LS) method. The ratio of computation ti...

Convex representations of shapes have several nice properties that can be exploited to generate efficient geometric algorithms. At the same time, extending algorithms from convex to non-convex shapes is non-trivial and often leads to more expensive solutions. An alternative and sometimes more efficient solution is to transform the non-convex problem into a collection of convex problems using a ...

In this paper, we develop a convexification tool that enables the construction of convex hulls for orthogonal disjunctive sets using convex extensions and disjunctive programming techniques. A distinguishing feature of our technique is that, unlike most applications of disjunctive programming, it does not require the introduction of new variables in the relaxation. We develop and apply a toolbo...

Proximal bundle methods have been shown to be highly successful optimization methods for unconstrained convex problems with discontinuous first derivatives. This naturally leads to the question of whether proximal variants of bundle methods can be extended to a nonconvex setting. This work proposes an approach based on generating cutting-planes models, not of the objective function as most bund...

In the most described maximum power point tracking (MPPT) methods in the literatures, the optimal operation point of the photovoltaic (PV) systems is estimated by linear approximations. However, these approximations can lead to less optimal operating conditions and significantly reduce the performances of the PV systems. This paper proposes a new approach to determine the maximum power point (M...

Christian Kirches develops a fast numerical algorithm of wide applicability that efficiently solves mixed-integer nonlinear optimal control problems. He uses convexification and relaxation techniques to obtain computationally tractable reformulations for which feasibility and optimality certificates...

In this contribution we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, e.g., from the outer convexification of integer control decisions. We treat this optimal control problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved using sequen...

We develop a framework for convexifying a fairly general class of optimization problems. Under additional assumptions, we analyze the suboptimality of the solution to the convexified problem relative to the original nonconvex problem and prove additive approximation guarantees. We then develop algorithms based on stochastic gradient methods to solve the resulting optimization problems and show ...