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In this paper, we discuss adaptive approximations of an elliptic eigenvalue optimization problem in a phase-field setting by a conforming finite element method. An adaptive algorithm is proposed and implemented in several two dimensional…
In this paper a spline based integral approximation is utilized to propose a sequence of approximations to the error function that converge at a significantly faster manner than the default Taylor series. The approximations can be improved…
A predominant topic in the theory of evolutionary algorithms and, more generally, theory of randomized black-box optimization techniques is running time analysis. Running time analysis aims at understanding the performance of a given…
Deep neural networks proved to be a very useful and powerful tool with many practical applications. They especially excel at learning from large data sets with labeled samples. However, in order to achieve good learning results, the network…
We survey key techniques and results from approximation theory in the context of uniform approximations to real functions such as e^{-x}, 1/x, and x^k. We then present a selection of results demonstrating how such approximations can be used…
In real-world applications, many optimization problems have the time-linkage property, that is, the objective function value relies on the current solution as well as the historical solutions. Although the rigorous theoretical analysis on…
Finding a \emph{single} best solution is the most common objective in combinatorial optimization problems. However, such a single solution may not be applicable to real-world problems as objective functions and constraints are only…
Solving an optimization task in any domain is a very challenging problem, especially when dealing with nonlinear problems and non-convex functions. Many meta-heuristic algorithms are very efficient when solving nonlinear functions. A…
Chance-constrained problems involve stochastic components in the constraints which can be violated with a small probability. We investigate the impact of different types of chance constraints on the performance of iterative search…
Evolutionary processes proved very useful for solving optimization problems. In this work, we build a formalization of the notion of cooperation and competition of multiple systems working toward a common optimization goal of the population…
Genetic fitness optimization using small populations or small population updates across generations generally suffers from randomly diverging evolutions. We propose a notion of highly probable fitness optimization through feasible…
Several classical adaptive optimization algorithms, such as line search and trust region methods, have been recently extended to stochastic settings where function values, gradients, and Hessians in some cases, are estimated via stochastic…
A common assumption in evolutionary thought is that adaptation drives an increase in biological complexity. However, the rules governing evolution of complexity appear more nuanced. Evolution is deeply connected to learning, where…
We propose a new way to self-adjust the mutation rate in population-based evolutionary algorithms in discrete search spaces. Roughly speaking, it consists of creating half the offspring with a mutation rate that is twice the current…
We consider approximation or recovery of functions based on a finite number of function evaluations. This is a well-studied problem in optimal recovery, machine learning, and numerical analysis in general, but many fundamental insights were…
Evolutionary algorithms (EA), a class of stochastic search methods based on the principles of natural evolution, have received widespread acclaim for their exceptional performance in various real-world optimization problems. While…
Recently, it has been proven that evolutionary algorithms produce good results for a wide range of combinatorial optimization problems. Some of the considered problems are tackled by evolutionary algorithms that use a representation which…
We describe a framework in which is possible to develop and implement algorithms for the approximation of invariant measures of dynamical systems with a given bound on the error of the approximation. Our approach is based on a general…
The predictive quality of machine learning models is typically measured in terms of their (approximate) expected prediction accuracy or the so-called Area Under the Curve (AUC). Minimizing the reciprocals of these measures are the goals of…
Modern large-scale statistical models require to estimate thousands to millions of parameters. This is often accomplished by iterative algorithms such as gradient descent, projected gradient descent or their accelerated versions. What are…