A set of quadratic forms is simultaneously diagonalizable via congruence (SDC) if there exists a basis under which each of the quadratic forms is diagonal. This property appears naturally when analyzing quadratically constrained quadratic programs (QCQPs) and has important implications in this context. This paper extends the reach of the SDC property by studying two … Read more
We describe a matrix-free trust-region algorithm for solving convex-constrained optimization problems that uses the spectral projected gradient method to compute trial steps. To project onto the intersection of the feasible set and the trust region, we reformulate and solve the dual projection problem as a one-dimensional root finding problem. We demonstrate our algorithm’s performance on … Read more
We present a new algorithm for infinite-dimensional optimization with general constraints, called ALESQP. In short, ALESQP is an augmented Lagrangian method that penalizes inequality constraints and solves equality-constrained nonlinear optimization subproblems at every iteration. The subproblems are solved using a matrix-free trust-region sequential quadratic programming (SQP) method that takes advantage of iterative, i.e., inexact linear … Read more
We study solution sensitivity for nonlinear programs (NLPs) whose structure is induced by a graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$. These graph-structured NLPs arise in many applications such as dynamic optimization, stochastic optimization, optimization with partial differential equations, and network optimization. We show that the sensitivity of the primal-dual solution at node $i\in \mathcal{V}$ against a data perturbation at … Read more
We study a feasibility-seeking problem with percentage violation constraints. These are additional constraints, that are appended to an existing family of constraints, which single out certain subsets of the existing constraints and declare that up to a specified fraction of the number of constraints in each subset is allowed to be violated by up to … Read more
There has been a long history of using ordinary differential equations (ODEs) to understand the dynamic of discrete-time algorithms (DTAs). Surprisingly, there are still two fundamental and unanswered questions: (i) it is unclear how to obtain a \emph{suitable} ODE from a given DTA, and (ii) it is unclear the connection between the convergence of a … Read more
Inspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis … Read more
The motivation for this paper stems from the desire to develop an adaptive sampling method for solving constrained optimization problems in which the objective function is stochastic and the constraints are deterministic. The method proposed in this paper is a proximal gradient method that can also be applied to the composite optimization problem min f(x) … Read more
The constraint nondegeneracy condition is one of the most relevant and useful constraint qualifications in nonlinear semidefinite programming. It can be characterized in terms of any fixed orthonormal basis of the, let us say, $\ell$-dimensional kernel of the constraint matrix, by the linear independence of a set of $\ell(\ell+1)/2$ derivative vectors. We show that this … Read more
We study the minimization of a rank-one quadratic with indicators and show that the underlying set function obtained by projecting out the continuous variables is supermodular. Although supermodular minimization is, in general, difficult, the specific set function for the rank-one quadratic can be minimized in linear time. We show that the convex hull of the … Read more