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We prove that linearizing certain families of polynomial optimization problems leads to new functorial operations in real convex sets. We show that under some conditions these operations can be computed or approximated in ways amenable to…
The problem of selecting optimal backdoor adjustment sets to estimate causal effects in graphical models with hidden and conditioned variables is addressed. Previous work has defined optimality as achieving the smallest asymptotic…
In this paper, we discuss scalar Lagrangian multipliers and vector Lagrangian multipliers for constrained set-valued optimization problems. We obtain some necessary conditions, sufficient conditions, as well as necessary and sufficient…
For optimization models to be used in practice, it is crucial that users trust the results. A key factor in this aspect is the interpretability of the solution process. A previous framework for inherently interpretable optimization models…
Subshifts are sets of colorings of $\mathbb{Z}^d$ defined by families of forbidden patterns. In a given subshift, the extender set of a finite pattern is the set of all its admissible completions. Since soficity of $\mathbb{Z}$ subshifts is…
Abductive explanations (AXp's) are widely used for understanding decisions of classifiers. Existing definitions are suitable when features are independent. However, we show that ignoring constraints when they exist between features may lead…
We derive two types of representation results for increasing convex functionals in terms of countably additive measures. The first is a max-representation of functionals defined on spaces of real-valued continuous functions and the second a…
The method of exhaustion is generalized to a simple formula that can be used to integrate functions under very general conditions, provided that the integral exists. Both a geometric proof (following the usual procedure for the method of…
Choice functions over a set $X$ that satisfy the Outcast, a.k.a. Aizerman, property are exactly those that attach to any set its maximal subset relative to some total order of ${2}^{X}$.
The class of uniformly computable real functions with respect to a small subrecursive class of operators computes the elementary functions of calculus, restricted to compact subsets of their domains. The class of conditionally computable…
We give new computable necessary conditions for a class of optimal transportation problems to have smooth solutions.
We consider a stochastic control problem where the set of strict (classical) controls is not necessarily convex, and the system is governed by a nonlinear backward stochastic differential equation. By introducing a new approach, we…
A class of parametric functions formed by alternating compositions of multivariate polynomials and rectification style monomial maps is studied (the layer-wise exponents are treated as fixed hyperparameters and are not optimized). For this…
Optimal power flow problems (OPFs) are mathematical programs used to determine how to distribute power over networks subject to network operation constraints and the physics of power flows. In this work, we take the view of treating an OPF…
We address the problem of obtaining well-defined criteria for multiobjective optimal control systems. Necessary and sufficient conditions for an optimal control functional to be nonessential are proved. The results provide effective tools…
Optimality Theory is a constraint-based theory of phonology which allows constraints to be violated. Consequently, implementing the theory presents problems for declarative constraint-based processing frameworks. On the basis of two…
Exposed positive maps in matrix algebras define a dense subset of extremal maps. We provide a sufficient condition for a positive map to be exposed. This is an analog of a spanning property which guaranties that a positive map is optimal.…
We find asymptotically optimal methods of recovery of the integration operator given values of the function at a finite number of points for a class of multivariate functions defined on a bounded star domain that have bounded in $L_p$ norm…
Motivated by the control of invasive biological populations, we consider a class of optimization problems for moving sets $t\mapsto \Omega(t)\subset\mathbb{R}^2$. Given an initial set $\Omega_0$, the goal is to minimize the area of the…
In this paper, we propose a new Fully Composite Formulation of convex optimization problems. It includes, as a particular case, the problems with functional constraints, max-type minimization problems, and problems of Composite…