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Systems of wave equations may fail to be globally well posed, even for small initial data. Attempts to classify systems into well and ill-posed categories work by identifying structural properties of the equations that can work as…
Inspired by classical sensitivity results for nonlinear optimization, we derive and discuss new quantitative bounds to characterize the solution map and dual variables of a parametrized nonlinear program. In particular, we derive explicit…
We leverage the connections between nonexpansive maps, monotone Lipschitz operators, and proximal mappings to obtain near-optimal (i.e., optimal up to poly-log factors in terms of iteration complexity) and parameter-free methods for solving…
An interesting observation is that most pairs of weakly homogeneous mappings have no strongly monotonic property, which is one of the key conditions to ensure the unique solvability of the generalized variational inequality. This paper…
Recent work on Bayesian optimization has shown its effectiveness in global optimization of difficult black-box objective functions. Many real-world optimization problems of interest also have constraints which are unknown a priori. In this…
Composite minimization is a powerful framework in large-scale convex optimization, based on decoupling of the objective function into terms with structurally different properties and allowing for more flexible algorithmic design. We…
We provide general formulation of weak identification in semiparametric models and an efficiency concept. Weak identification occurs when a parameter is weakly regular, i.e., when it is locally homogeneous of degree zero. When this happens,…
Regularization, whether explicit in terms of a penalty in the loss or implicit in the choice of algorithm, is a cornerstone of modern machine learning. Indeed, controlling the complexity of the model class is particularly important when…
We present Nonstandard Analysis by three axioms: the {\em Extension, Transfer and Saturation Principles} in the framework of the superstructure of a given infinite set. We also present several applications of this axiomatic approach to…
Regularization and interior point approaches offer valuable perspectives to address constrained nonlinear optimization problems in view of control applications. This paper discusses the interactions between these techniques and proposes an…
Based on the tools of limiting variational analysis, we derive a sequential necessary optimality condition for nonsmooth mathematical programs which holds without any additional assumptions. In order to ensure that stationary points in this…
We present a method to solve a special class of parameter identification problems for an elliptic optimal control problem to global optimality. The bilevel problem is reformulated via the optimal-value function of the lower-level problem.…
Many recent studies on first-order methods (FOMs) focus on \emph{composite non-convex non-smooth} optimization with linear and/or nonlinear function constraints. Upper (or worst-case) complexity bounds have been established for these…
This contribution examines optimization problems that involve stochastic dominance constraints. These problems have uncountably many constraints. We develop methods to solve the optimization problem by reducing the constraints to a finite…
Methods of Lie group analysis of differential equations are extended to weak solutions of (linear and nonlinear) PDEs, where the term ``weak solution'' comprises the following settings: (a) Distributional solutions. (b) Solutions in…
In this paper, we develop an analytical framework for the partial differential equation underlying the consensus-based optimization model. The main challenge arises from the nonlinear, nonlocal nature of the consensus point, coupled with a…
The main focus of these notes is recent work on linear systems in which line arrangements play a role, including problems such as semi-effectivity, containment problems of symbolic powers of homogeneous ideals in their powers, bounded…
In a decision-theoretic framework, the minimax lower bound provides the worst-case performance of estimators relative to a given class of statistical models. For parametric and semiparametric models, the H\'{a}jek--Le Cam local asymptotic…
This paper investigates the optimality conditions for characterizing the local minimizers of the constrained optimization problems involving an $\ell_p$ norm ($0<p<1$) of the variables, which may appear in either the objective or the…
We study the performance of asymptotic and approximate consensus algorithms under harsh environmental conditions. The asymptotic consensus problem requires a set of agents to repeatedly set their outputs such that the outputs converge to a…