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We study the computational complexity of multi-stage robust optimization problems. Such problems are formulated with alternating min/max quantifiers and therefore naturally fall into a higher stage of the polynomial hierarchy. Despite this, almost no hardness results with respect to the polynomial hierarchy are known. In this work, we examine the hardness of robust two-stage adjustable and robust recoverable optimization with budgeted uncertainty sets. Our main technical contribution is the introduction of a technique tailored to prove $\Sigma^p_3$-hardness of such problems. We highlight a difference between continuous and discrete budgeted uncertainty: In the discrete case, indeed a wide range of problems becomes complete for the third stage of the polynomial hierarchy; in particular, this applies to the TSP, independent set, and vertex cover problems. However, in the continuous case this does not happen and problems remain in the first stage of the hierarchy. Finally, if we allow the uncertainty to not only affect the objective, but also multiple constraints, then this distinction disappears and even in the continuous case we encounter hardness for the third stage of the hierarchy. This shows that even robust problems which are already NP-complete can still exhibit a significant computational difference between column-wise and row-wise uncertainty.

Estimation-of-distribution algorithms (EDAs) are optimization algorithms that learn a distribution on the search space from which good solutions can be sampled easily. A key parameter of most EDAs is the sample size (population size). If the population size is too small, the update of the probabilistic model builds on few samples, leading to the undesired effect of genetic drift. Too large population sizes avoid genetic drift, but slow down the process. Building on a recent quantitative analysis of how the population size leads to genetic drift, we design a smart-restart mechanism for EDAs. By stopping runs when the risk for genetic drift is high, it automatically runs the EDA in good parameter regimes. Via a mathematical runtime analysis, we prove a general performance guarantee for this smart-restart scheme. This in particular shows that in many situations where the optimal (problem-specific) parameter values are known, the restart scheme automatically finds these, leading to the asymptotically optimal performance. We also conduct an extensive experimental analysis. On four classic benchmark problems, we clearly observe the critical influence of the population size on the performance, and we find that the smart-restart scheme leads to a performance close to the one obtainable with optimal parameter values. Our results also show that previous theory-based suggestions for the optimal population size can be far from the optimal ones, leading to a performance clearly inferior to the one obtained via the smart-restart scheme. We also conduct experiments with PBIL (cross-entropy algorithm) on two combinatorial optimization problems from the literature, the max-cut problem and the bipartition problem. Again, we observe that the smart-restart mechanism finds much better values for the population size than those suggested in the literature, leading to a much better performance.

ChatGPT is a large language model trained by OpenAI. In this technical report, we explore for the first time the capability of ChatGPT for programming numerical algorithms. Specifically, we examine the capability of GhatGPT for generating codes for numerical algorithms in different programming languages, for debugging and improving written codes by users, for completing missed parts of numerical codes, rewriting available codes in other programming languages, and for parallelizing serial codes. Additionally, we assess if ChatGPT can recognize if given codes are written by humans or machines. To reach this goal, we consider a variety of mathematical problems such as the Poisson equation, the diffusion equation, the incompressible Navier-Stokes equations, compressible inviscid flow, eigenvalue problems, solving linear systems of equations, storing sparse matrices, etc. Furthermore, we exemplify scientific machine learning such as physics-informed neural networks and convolutional neural networks with applications to computational physics. Through these examples, we investigate the successes, failures, and challenges of ChatGPT. Examples of failures are producing singular matrices, operations on arrays with incompatible sizes, programming interruption for relatively long codes, etc. Our outcomes suggest that ChatGPT can successfully program numerical algorithms in different programming languages, but certain limitations and challenges exist that require further improvement of this machine learning model.

Satisfiability Modulo the Theory of Nonlinear Real Arithmetic, SMT(NRA) for short, concerns the satisfiability of polynomial formulas, which are quantifier-free Boolean combinations of polynomial equations and inequalities with integer coefficients and real variables. In this paper, we propose a local search algorithm for a special subclass of SMT(NRA), where all constraints are strict inequalities. An important fact is that, given a polynomial formula with $n$ variables, the zero level set of the polynomials in the formula decomposes the $n$-dimensional real space into finitely many components (cells) and every polynomial has constant sign in each cell. The key point of our algorithm is a new operation based on real root isolation, called cell-jump, which updates the current assignment along a given direction such that the assignment can `jump' from one cell to another. One cell-jump may adjust the values of several variables while traditional local search operations, such as flip for SAT and critical move for SMT(LIA), only change that of one variable. We also design a two-level operation selection to balance the success rate and efficiency. Furthermore, our algorithm can be easily generalized to a wider subclass of SMT(NRA) where polynomial equations linear with respect to some variable are allowed. Experiments show the algorithm is competitive with state-of-the-art SMT solvers, and performs particularly well on those formulas with high-degree polynomials.

We develop in this work the first polytopal complexes of differential forms. These complexes, inspired by the Discrete De Rham and the Virtual Element approaches, are discrete versions of the de Rham complex of differential forms built on meshes made of general polytopal elements. Both constructions benefit from the high-level approach of polytopal methods, which leads, on certain meshes, to leaner constructions than the finite element method. We establish commutation properties between the interpolators and the discrete and continuous exterior derivatives, prove key polynomial consistency results for the complexes, and show that their cohomologies are isomorphic to the cohomology of the continuous de Rham complex.

Likelihood-based inferences have been remarkably successful in wide-spanning application areas. However, even after due diligence in selecting a good model for the data at hand, there is inevitably some amount of model misspecification: outliers, data contamination or inappropriate parametric assumptions such as Gaussianity mean that most models are at best rough approximations of reality. A significant practical concern is that for certain inferences, even small amounts of model misspecification may have a substantial impact; a problem we refer to as brittleness. This article attempts to address the brittleness problem in likelihood-based inferences by choosing the most model friendly data generating process in a discrepancy-based neighbourhood of the empirical measure. This leads to a new Optimistically Weighted Likelihood (OWL), which robustifies the original likelihood by formally accounting for a small amount of model misspecification. Focusing on total variation (TV) neighborhoods, we study theoretical properties, develop inference algorithms and illustrate the methodology in applications to mixture models and regression.

Technology adoption research aims to determine the reasons why and how individuals, corporations, and industries start using new technology. Furthermore, technology adoption itself is decomposed into underlying sub-processes which are characterized by a finite number of sequential states in order to capture its evolutionary nature. Building upon that, in this paper a technology adoption index is being constructed that allows for statistical testing. This new framework is flexible with respect to the number of underlying models, and accounts for nonlinearities within the evolution of technology adoption. It can be considered as novel because it gives opportunity to a quantitative analysis of technology adoption that has not existed before. Subsequently, this framework is applied for an integrated model of technology adoption.

Quadratization of polynomial and nonpolynomial systems of ordinary differential equations is advantageous in a variety of disciplines, such as systems theory, fluid mechanics, chemical reaction modeling and mathematical analysis. A quadratization reveals new variables and structures of a model, which may be easier to analyze, simulate, control, and provides a convenient parametrization for learning. This paper presents novel theory, algorithms and software capabilities for quadratization of non-autonomous ODEs. We provide existence results, depending on the regularity of the input function, for cases when a quadratic-bilinear system can be obtained through quadratization. We further develop existence results and an algorithm that generalizes the process of quadratization for systems with arbitrary dimension that retain the nonlinear structure when the dimension grows. For such systems, we provide dimension-agnostic quadratization. An example is semi-discretized PDEs, where the nonlinear terms remain symbolically identical when the discretization size increases. As an important aspect for practical adoption of this research, we extended the capabilities of the QBee software towards both non-autonomous systems of ODEs and ODEs with arbitrary dimension. We present several examples of ODEs that were previously reported in the literature, and where our new algorithms find quadratized ODE systems with lower dimension than the previously reported lifting transformations. We further highlight an important area of quadratization: reduced-order model learning. This area can benefit significantly from working in the optimal lifting variables, where quadratic models provide a direct parametrization of the model that also avoids additional hyperreduction for the nonlinear terms. A solar wind example highlights these advantages.

This paper is concerned with the sample efficiency of reinforcement learning, assuming access to a generative model (or simulator). We first consider $\gamma$-discounted infinite-horizon Markov decision processes (MDPs) with state space $\mathcal{S}$ and action space $\mathcal{A}$. Despite a number of prior works tackling this problem, a complete picture of the trade-offs between sample complexity and statistical accuracy is yet to be determined. In particular, all prior results suffer from a severe sample size barrier, in the sense that their claimed statistical guarantees hold only when the sample size exceeds at least $\frac{|\mathcal{S}||\mathcal{A}|}{(1-\gamma)^2}$. The current paper overcomes this barrier by certifying the minimax optimality of two algorithms -- a perturbed model-based algorithm and a conservative model-based algorithm -- as soon as the sample size exceeds the order of $\frac{|\mathcal{S}||\mathcal{A}|}{1-\gamma}$ (modulo some log factor). Moving beyond infinite-horizon MDPs, we further study time-inhomogeneous finite-horizon MDPs, and prove that a plain model-based planning algorithm suffices to achieve minimax-optimal sample complexity given any target accuracy level. To the best of our knowledge, this work delivers the first minimax-optimal guarantees that accommodate the entire range of sample sizes (beyond which finding a meaningful policy is information theoretically infeasible).