In this paper, we focus our study on the convergence of (proximal) gradient methods and accelerated (proximal) gradient methods for smooth (composite) optimization under a H\”{o}lderian error bound (HEB) condition. We first show that proximal gradient (PG) method is automatically adaptive to HEB while accelerated proximal gradient (APG) method can be adaptive to HEB by … Read more
Representability results play a fundamental role in optimization since they provide characterizations of the feasible sets that arise from optimization problems. In this paper we study the sets that appear in the feasibility version of mixed binary convex quadratic optimization problems. We provide a complete characterization of the sets that can be obtained as the … Read more
PDE-constrained optimization aims at finding optimal setups for partial differential equations so that relevant quantities are minimized. Including sparsity promoting terms in the formulation of such problems results in more practically relevant computed controls but adds more challenges to the numerical solution of these problems. The needed L^1-terms as well as additional inclusion of box … Read more
We are interested in the existence of Pareto solutions to the vector optimization problem $$\text{\rm Min}_{\,\mathbb{R}^m_+} \{f(x) \,|\, x\in \mathbb{R}^n\},$$ where $f\colon\mathbb{R}^n\to \mathbb{R}^m$ is a polynomial map. By using the {\em tangency variety} of $f$ we first construct a semi-algebraic set of dimension at most $m – 1$ containing the set of Pareto values of … Read more
We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory–Johnson model. These notions were known to coincide for continuous piecewise linear functions with rational breakpoints. We show that two of the notions, extreme functions and facets, coincide for the case of continuous piecewise linear functions, removing … Read more
We show how random projections can be used to solve large-scale dense linear programs approximately. This is a new application of techniques which are now fairly well known in probabilistic algorithms, but have never yet been systematically applied to the fundamental class of Linear Programming. We develop the necessary theoretical framework, and show that this … Read more
We contribute improvements to a Lagrangian dual solution approach applied to large-scale optimization problems whose objective functions are convex, continuously differentiable and possibly nonlinear, while the non-relaxed constraint set is compact but not necessarily convex. Such problems arise, for example, in the split-variable deterministic reformulation of stochastic mixed-integer optimization problems. The dual solution approach needs … Read more
Electronic structure calculations, in particular the computation of the ground state energy, lead to challenging problems in optimization. These problems are of enormous importance in quantum chemistry for calculations of properties of solids and molecules. Minimization methods for computing the ground state energy can be developed by employing a variational approach, where the second-order reduced … Read more
The Mesh Adaptive Direct Search algorithm (MADS) is an iterative method for constrained blackbox optimization problems. One of the optional MADS features is a versatile search step in which quadratic models are built leading to a series of quadratically constrained quadratic subproblems. This work explores different algorithms that exploit the structure of the quadratic models: … Read more
We study statistical properties of the optimal value and optimal solutions of the Sample Average Approximation of risk averse stochastic problems. Central Limit Theorem type results are derived for the optimal value when the stochastic program is expressed in terms of a law invariant coherent risk measure having a discrete Kusuoka representation. The obtained results … Read more