We introduce a distributionally robust minimium mean square error estimation model with a Wasserstein ambiguity set to recover an unknown signal from a noisy observation. The proposed model can be viewed as a zero-sum game between a statistician choosing an estimator—that is, a measurable function of the observation—and a fictitious adversary choosing a prior—that is, … Read more
Several logarithmic-barrier interior-point methods are now available for solving two-stage stochastic optimization problems with recourse in the finite-dimensional setting. However, despite the genuine need for studying such methods in general spaces, there are no infinite-dimensional analogs of these methods. Inspired by this evident gap in the literature, in this paper, we propose logarithmic-barrier decomposition-based interior-point … Read more
In this paper we propose a splitting scheme which hybridizes generalized conditional gradient with a proximal step which we call CGALP algorithm, for minimizing the sum of three proper convex and lower-semicontinuous functions in real Hilbert spaces. The minimization is subject to an affine constraint, that allows in particular to deal with composite problems (sum … Read more
We consider the semi-infinite system of polynomial inequalities of the form \[ \mathbf{K}:=\{x\in\mathbb{R}^m\mid p(x,y)\ge 0,\ \ \forall y\in S\subseteq\mathbb{R}^n\}, \] where $p(X,Y)$ is a real polynomial in the variables $X$ and the parameters $Y$, the index set $S$ is a basic semialgebraic set in $\mathbb{R}^n$, $-p(X,y)$ is convex in $X$ for every $y\in S$. We … Read more
In this paper, we study the two-stage stochastic linear semi-infinite programming with recourse to handle uncertainty in data defining (deterministic) linear semi-infinite programming. We develop and analyze volumetric barrier decomposition-based interior point methods for solving this class of optimization problems, and present a complexity analysis of the proposed algorithms. We establish our convergence analysis by … Read more
We present a novel direct transcription method to solve optimization problems subject to nonlinear differential and inequality constraints. In order to provide numerical convergence guarantees, it is sufficient for the functions that define the problem to satisfy boundedness and Lipschitz conditions. Our assumptions are the most general to date; we do not require uniqueness, differentiability … Read more
We point out that Chubanov’s oracle-based algorithm for linear programming [5] can be applied almost as it is to linear semi-infinite programming (LSIP). In this note, we describe the details and prove the polynomial complexity of the algorithm based on the real computation model proposed by Blum, Shub and Smale (the BSS model) which is … Read more
We study time-varying semidefinite programs (TV-SDPs), which are semidefinite programs whose data (and solutions) are functions of time. Our focus is on the setting where the data varies polynomially with time. We show that under a strict feasibility assumption, restricting the solutions to also be polynomial functions of time does not change the optimal value … Read more
A robust-to-dynamics optimization (RDO) problem} is an optimization problem specified by two pieces of input: (i) a mathematical program (an objective function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ and a feasible set $\Omega\subseteq\mathbb{R}^n$), and (ii) a dynamical system (a map $g:\mathbb{R}^n\rightarrow\mathbb{R}^n$). Its goal is to minimize $f$ over the set $\mathcal{S}\subseteq\Omega$ of initial conditions that forever remain in $\Omega$ under … Read more
This work presents a convex-optimization-based framework for analysis and control of nonlinear partial differential equations. The approach uses a particular weak embedding of the nonlinear PDE, resulting in a \emph{linear} equation in the space of Borel measures. This equation is then used as a constraint of an infinite-dimensional linear programming problem (LP). This LP is … Read more