相关论文: A geometry in the set of solutions to ill-posed li…
Consider the linear equation $\mathbf{A}\mathbf{x}=\mathbf{y}$, where $\mathbf{A}$ is a $k\times N$-matrix, $\mathbf{x}\in\mathcal{K}\subset \mathbb{R}^N$ and $\mathbf{y}\in\mathbb{R}^M$ a given vector. When $\mathcal{K}$ is a convex set…
We study computational and statistical consequences of problem geometry in stochastic and online optimization. By focusing on constraint set and gradient geometry, we characterize the problem families for which stochastic- and…
Constrained optimization problems exist in many domains of science, such as thermodynamics, mechanics, economics, etc. These problems are classically solved with the help of the Lagrange multipliers and the Lagrangian function. However, the…
Entropy-based (M_N) moment closures for kinetic equations are defined by a constrained optimization problem that must be solved at every point in a space-time mesh, making it important to solve these optimization problems accurately and…
We present a geometric multilevel optimization approach that smoothly incorporates box constraints. Given a box constrained optimization problem, we consider a hierarchy of models with varying discretization levels. Finer models are…
We introduce an algorithm which can be directly used to feasible and optimum search in linear programming. Starting from an initial point the algorithm iteratively moves a point in a direction to resolve the violated constraints. At the…
This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints…
We study entropy-bounded computational geometry, that is, geometric algorithms whose running times depend on a given measure of the input entropy. Specifically, we introduce a measure that we call range-partition entropy, which unifies and…
We introduce a general random model of a combinatorial optimization problem with geometric structure that encapsulates both linear programming and integer linear programming. Let $Q$ be a bounded set called the feasible set, $E$ be an…
For a large number of random constraint satisfaction problems, such as random k-SAT and random graph and hypergraph coloring, there are very good estimates of the largest constraint density for which solutions exist. Yet, all known…
We review several (and provide new) results on the theory of moments, sums of squares and basic semi-algebraic sets when convexity is present. In particular, we show that under convexity, the hierarchy of semidefinite relaxations for…
In this work, we focus on separable convex optimization problems with box constraints and a set of triangular linear constraints. The solution is given in closed-form as a function of some Lagrange multipliers that can be computed through…
We present a theoretical framework for characterizing the geometrical properties of the space of solutions in constraint satisfaction problems, together with practical algorithms for studying this structure on particular instances. We apply…
This paper investigates the geometric constraints imposed on a domain by overdetermined problems for partial differential equations. Serrin's symmetry results are extended to overdetermined problems with potentially degenerate ellipticity…
In this paper, we are concerned with geometric constraint solvers, i.e., with programs that find one or more solutions of a geometric constraint problem. If no solution exists, the solver is expected to announce that no solution has been…
This paper focusses on the optimal control problems governed by fourth-order linear elliptic equations with clamped boundary conditions in the framework of the Hessian discretisation method (HDM). The HDM is an abstract framework that…
We solve several problems that involve imposing metrics on surfaces. The problem of a strip with a linear metric gradient is formulated in terms of a Lagrangean similar to those used for spin systems. We are able to show that the low energy…
Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…
In this work, we focus on separable convex optimization problems with linear and box constraints and compute the solution in closed-form as a function of some Lagrange multipliers that can be easily computed in a finite number of…
We review the properties of the constraint equations, from their geometric origin in hypersurface geometry through to their roles in the Cauchy problem and the Hamiltonian formulation of the Einstein equations. We then review properties of…