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The ability to differentiate through optimization problems has unlocked numerous applications, from optimization-based layers in machine learning models to complex design problems formulated as bilevel programs. It has been shown that…
Partial differential equation (PDE)-constrained optimization, where an optimization problem is subject to PDE constraints, arises in various applications such as design, control, and inference. Solving such problems is computationally…
In this paper, we propose some new semidefinite relaxations for a class of nonconvex complex quadratic programming problems, which widely appear in the areas of signal processing and power system. By deriving new valid constraints to the…
We consider the problem of finding the optimal diagonal preconditioner for a positive definite matrix. Although this problem has been shown to be solvable and various methods have been proposed, none of the existing approaches are scalable…
The main focus of this paper is the characterization and exploitation of the asymptotic spectrum of the saddle--point matrix sequences arising from the discretization of optimization problems constrained by elliptic partial differential…
This article introduces a general purpose framework and software to approximate partial differential equations (PDEs). The sparsity patterns of finite element discretized operators is identified automatically using the tools from…
We introduce a new algorithm and software for solving linear equations in symmetric diagonally dominant matrices with non-positive off-diagonal entries (SDDM matrices), including Laplacian matrices. We use pre-conditioned conjugate gradient…
A class of quasilinear singularly perturbed boundary value problems with a turning point of attractive type is considered. The problems are solved numerically by a finite-difference scheme on a special discretization mesh which is dense…
We design two classes of ultra-fast meta-solvers for linear systems arising after discretizing PDEs by combining neural operators with either simple iterative solvers, e.g., Jacobi and Gauss-Seidel, or with Krylov methods, e.g., GMRES and…
Seeking tighter relaxations of combinatorial optimization problems, semidefinite programming is a generalization of linear programming that offers better bounds and is still polynomially solvable. Yet, in practice, a semidefinite program is…
We develop two adaptive discretization algorithms for convex semi-infinite optimization, which terminate after finitely many iterations at approximate solutions of arbitrary precision. In particular, they terminate at a feasible point of…
We present a simple way to discretize and precondition mixed variational formulations. Our theory connects with, and takes advantage of, the classical theory of symmetric saddle point problems and the theory of preconditioning symmetric…
We develop efficient hierarchical preconditioners for optimal control problems governed by partial differential equations with uncertain coefficients. Adopting a discretize-then-optimize framework that integrates finite element…
Finite linear least squares is one of the core problems of numerical linear algebra, with countless applications across science and engineering. Consequently, there is a rich and ongoing literature on algorithms for solving linear least…
Semidefinite programs (SDPs) are standard convex problems that are frequently found in control and optimization applications. Interior-point methods can solve SDPs in polynomial time up to arbitrary accuracy, but scale poorly as the size of…
We present a novel variational quantum framework for nonlinear partial differential equation (PDE) constrained optimization problems. The proposed work extends the recently introduced bi-level variational quantum PDE constrained…
In linear optimization, matrix structure can often be exploited algorithmically. However, beneficial presolving reductions sometimes destroy the special structure of a given problem. In this article, we discuss structure-aware…
Incomplete factorizations have long been popular general-purpose algebraic preconditioners for solving large sparse linear systems of equations. Guaranteeing the factorization is breakdown free while computing a high quality preconditioner…
Discretizations of infinite-dimensional variational inequalities lead to linear and nonlinear complementarity problems with many degrees of freedom. To solve these problems in a parallel computing environment, we propose two active-set…
Mixed dimensional partial differential equations (PDEs) are equations coupling unknown fields defined over domains of differing topological dimension. Such equations naturally arise in a wide range of scientific fields including geology,…