We first study the fast minimization properties of the trajectories of the second-order evolution equation \begin{equation*} \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \beta \nabla^2 \Phi (x(t))\dot{x} (t) + \nabla \Phi (x(t)) = 0, \end{equation*} where $\Phi : \mathcal H \to \mathbb R$ is a smooth convex function acting on a real Hilbert space $\mathcal H$, and … Read more
Consumers are increasingly navigating across sales channels to make purchases. The common retail practice of pricing channels independently is unable to achieve the desired profitable coordination required between channels. As part of a joint partnership agreement with IBM Commerce, we engaged with three major retailers over two years, and developed advanced omni-channel pricing (OCP) solutions … Read more
Low-rank tensor methods provide efficient representations and computations for high-dimensional problems and are able to break the curse of dimensionality when dealing with systems involving multiple parameters. We present algorithms for constrained nonlinear optimization problems that use low-rank tensors and apply them to optimal control of PDEs with uncertain parameters and to parametrized variational inequalities. … Read more
We consider a model of a pay-as-bid electricity market based on a multi-leader-common-follower approach where the producers as leaders are at the upper level and the regulator as a common follower is at the lower level. We fully characterise Nash equilibria for this model by describing necessary and sufficient conditions for their existence as well … Read more
We consider a multi-leader-common-follower model of a pay-as-bid electricity market in which the producers provide the regulator with either linear or quadratic bids. We prove that for a given producer only linear bids can maximise his profit. Such linear bids are referred as the “best response” of the given producer. They are obtained assuming the … Read more
Building on the work of Gendreau, Laporte, and Seguin (1996), we review the past 20 years of scientific literature on stochastic vehicle routing problems (SVRP). The numerous variants of the problem that have been studied in the literature are described and categorized. Also a thorough review of solution methods applied to the SVRP is included … Read more
Let $W(A)$ denote the field of values (numerical range) of a matrix $A$. For any polynomial $p$ and matrix $A$, define the Crouzeix ratio to have numerator $\max\left\{|p(\zeta)|:\zeta\in W(A)\right\}$ and denominator $\|p(A)\|_2$. M.~Crouzeix’s 2004 conjecture postulates that the globally minimal value of the Crouzeix ratio is $1/2$, over all polynomials $p$ of any degree and … Read more
Investment decisions in renewable energy sources such as small hydro, wind power, biomass and solar are frequently made in the context of enormous uncertainty surrounding both intermittent generation and the highly volatile electricity spot prices that are used for clearing of trades. This paper presents a new portfolio-based approach for selecting long-term investments in small-scale … Read more
We consider multistage stochastic linear optimization problems combining joint dynamic probabilistic constraints with hard constraints. We develop a method for projecting decision rules onto hard constraints of wait-and-see type. We establish the relation between the original (infinite dimensional) problem and approximating problems working with projections from different subclasses of decision policies. Considering the subclass of … Read more
We consider a class of augmented Lagrangian methods for solving convex programming problems with inequality constraints. This class involves a family of penalty functions and specific values of parameters $p,q,\tilde y \in R$ and $c>0$. The penalty family includes the classical modified barrier and the exponential function. The associated proximal method for solving the dual … Read more