The Low Order-Value Optimization (LOVO) problem involves minimizing the minimum among a finite number of function values within a feasible set. LOVO has several practical applications such as robust parameter estimation, protein alignment, portfolio optimization, among others. In this work, we are interested in the constrained nonlinear optimization LOVO problem of minimizing the minimum between a finite number of function values subject to a nonempty closed convex set where each function is a black-box and continuously differentiable, but the derivatives are not available. We develop the first derivative-free trust-region algorithm for constrained LOVO problems with convergence to weakly critical points. Under suitable conditions, we establish the global convergence of the algorithm and also its worst-case iteration complexity analysis. An initial open-source implementation using only linear interpolation models is developed. Extensive numerical experiments and comparison with existing alternatives show the properties and the efficiency of the proposed approach when solving LOVO problems.
Automatic differentiation (the automatic computation of numerical function derivatives) has many applications, especially in scientific computing and machine learning. Several approaches to automate the calculation of derivatives have been presented. Common challenges facing those approaches include (1) having to define derivative rules for different classes of functions, (2) having to manipulate program symbols to understand the semantics of the function being differentiated, and construct the derivative function, and (3), more importantly, ensuring that the calculated derivative is correct. In this paper, we present an approach for automatic \emph{certified} differentiation using the Lean proof assistant. By certified, we mean a correctness certificate is computed together with the derivative function and a domain on which the derivative is guaranteed to be valid. Our approach leverages existing derivative theorems included in the Mathlib Lean library instead of having to define our own rules. We also utilize Lean metaprogramming to manipulate the symbolic structure of the input function, and to construct the derivative function. The correctness certificate is computed step-wise as we apply Mathlib theorems. To make the search efficient, we index the derivative theorems by the function they differentiate. We evaluate our approach on 500 randomly generated functions: with the index, all of them are differentiated, with a median time of 0.18\,s per function, 8.6 times faster than without the index.
Mathematical software has particular needs that are not all covered by the existing literature on software engineering or algorithm design. This paper looks at some of these aspects, in the concrete context of the computer algebra system GAP. It also studies the potential impact AI systems could have on the development process for mathematical software.
We describe a new statistical test for pseudorandom number generators (PRNGs). Our test can find bias induced by dependencies among the Hamming weights of the outputs of a PRNG, even for PRNGs that pass state-of-the-art tests of the same kind from the literature, and in particular for generators based on $\mathbf F_2$-linear transformations such as the dSFMT, xoroshiro1024+, and WELL512a.
Recovering the governing equations of a dynamical system from data is a central problem in data-driven science, yet the available Python tooling forces a compromise among automation, statistical rigor, and reliability. We present py-ARGOS, a Python package implementing two automatic algorithms, bootstrap ARGOS and Bayesian-ARGOS, behind a uniform interface. Both are statistically principled throughout: automated model selection reduces the candidate library, each coefficient is reported with a bootstrap confidence or Bayesian credible interval that quantifies its uncertainty, and a term is retained when this interval excludes zero. Built-in diagnostics additionally flag unreliable fits. On chaotic benchmark systems, the two algorithms improve on the SINDy family's identification performance.