In the classical best approximation pair (BAP) problem, one is given two nonempty, closed, convex and disjoint subsets in a finite- or an infinite-dimensional Hilbert space, and the goal is to find a pair of points, each from each subset, which realizes the distance between the subsets. We discuss the problem in more general normed spaces and with possibly non-convex subsets, and focus our attention on the issues of uniqueness and existence of the solution to the problem. As far as we know, these fundamental issues have not received much attention. We present several sufficient geometric conditions for the (at most) uniqueness of a BAP. These conditions are related to the structure and the relative orientation of the boundaries of the subsets and to the norm. We also present many sufficient conditions for the existence of a BAP. Our results significantly extend the horizon of a recent algorithm for solving the BAP problem [Censor, Mansour, Reem, J. Approx. Theory (2024)]. The paper also shows, perhaps for the first time, how wide is the scope of the BAP problem in terms of the scientific communities which are involved in it (frequently independently) and in terms of its applications.

This paper describes the comprehensive safety framework th at underpinned the development, release process, and regulatory approval of BMW's first SAE Level 3 Au tomated Driving System. The framework combines established qualitative and quantitative me thods from the fields of Systems Engineering, Engineering Risk Analysis, Bayesian Data Analysis, Design of Experiments, and Statistical Learning in a novel manner. The approach systematically minimizes the r isks associated with hardware and software faults, performance limitations, and insufficient specifica tions to an acceptable level that achieves a Positive Risk Balance. At the core of the framework is the system atic identification and quantification of uncertainties associated with hazard scenarios and the red undantly designed system based on designed experiments, field data, and expert knowledge. The residual risk of the system is then estimated through Stochastic Simulation and evaluated by Sensitivity Analys is. By integrating these advanced analytical techniques into the V-Model, the framework fulfills, unifies, and complements existing automotive safety standards. It therefore provides a comprehensive, rigorou s, and transparent safety assurance process for the development and deployment of Automated Driving System s.

This paper studies policy evaluation with multiple data sources, especially in scenarios that involve one experimental dataset with two arms, complemented by a historical dataset generated under a single control arm. We propose novel data integration methods that linearly integrate base policy value estimators constructed based on the experimental and historical data, with weights optimized to minimize the mean square error (MSE) of the resulting combined estimator. We further apply the pessimistic principle to obtain more robust estimators, and extend these developments to sequential decision making. Theoretically, we establish non-asymptotic error bounds for the MSEs of our proposed estimators, and derive their oracle, efficiency and robustness properties across a broad spectrum of reward shift scenarios. Numerical experiments and real-data-based analyses from a ridesharing company demonstrate the superior performance of the proposed estimators.

Optimal control problems subject to partial differential equations (PDE) constraints represent a very important class of problems in practice and have found various important applications in science and engineering, e.g., fluid flow, heat conduction, structural optimization, microelectronics, crystal growth, vascular surgery, and cardiac medicine, to name just a few. The mathematical theory of distributed / boundary optimal control problems subject to linear / nonlinear elliptic / parabolic PDEs is well understood (see, e.g., the monographs [27, 39, 30]). Numerically, one of the most common approaches to solve such problems utilizes the first-order optimality system and introduces either adjoint state or Lagrangian multiplier, and then iteratively updates the control using established optimization algorithms, e.g., (accelerated / stochastic) gradient descent method, semismooth Newton method, and alternating direction method of multipliers. In practical implementation, the resulting continuous formulation is then often discretized by finite element method (FEM) and spectral methods. In the past few years, deep neural networks (DNNs) have demonstrated remarkable performance across a wide range of applications, e.g., computer vision, imaging, and natural language processing. These successes have permeated into several scientific disciplines, and DNNs have also become very popular tools to approximate solutions to various PDEs. The ease of implementation, and the potential to overcome the notorious curse of dimensionality have sparked intensive interest in revisiting classical computational problems from a deep-learning perspective. In the last few years, a large number of neural solvers based on DNNs have been proposed for solving PDEs, e.g., physics informed neural network (PINN) [34], deep Ritz method [14], deep Galerkin method [36], weak adversarial networks [41], and deep least-squares method [9].

Bilevel optimization reveals the inner structure of otherwise oblique optimization problems, such as hyperparameter tuning and meta-learning. A common goal in bilevel optimization is to find stationary points of the hyper-objective function. Although this hyper-objective approach is widely used, its theoretical properties have not been thoroughly investigated in cases where the lower-level functions lack strong convexity. In this work, we take a step forward and study the hyper-objective approach without the typical lower-level strong convexity assumption. Our hardness results show that the hyper-objective of general convex lower-level functions can be intractable either to evaluate or to optimize. To tackle this challenge, we introduce the gradient dominant condition, which strictly relaxes the strong convexity assumption by allowing the lower-level solution set to be non-singleton. Under the gradient dominant condition, we propose the Inexact Gradient-Free Method (IGFM), which uses the Switching Gradient Method (SGM) as the zeroth order oracle, to find an approximate stationary point of the hyper-objective. We also extend our results to nonsmooth lower-level functions under the weak sharp minimum condition.

It is widely acknowledged that transparency of automated decision making is crucial for deployability of intelligent systems, and explaining the reasons why some decisions are "good" and some are not is a way to achieving this transparency. We consider two variants of decision making, where "good" decisions amount to alternatives (i) meeting "most" goals, and (ii) meeting "most preferred" goals. We then define, for each variant and notion of "goodness" (corresponding to a number of existing notions in the literature), explanations in two formats, for justifying the selection of an alternative to audiences with differing needs and competences: lean explanations, in terms of goals satisfied and, for some notions of "goodness", alternative decisions, and argumentative explanations, reflecting the decision process leading to the selection, while corresponding to the lean explanations. To define argumentative explanations, we use assumption-based argumentation (ABA), a well-known form of structured argumentation. Specifically, we define ABA frameworks such that "good" decisions are admissible ABA arguments and draw argumentative explanations from dispute trees sanctioning this admissibility. Finally, we instantiate our overall framework for explainable decision-making to accommodate connections between goals and decisions in terms of decision graphs incorporating defeasible and non-defeasible information.

Possibilistic logic is the most popular approach to represent and reason with uncertain and partially inconsistent knowledge. Regarding normal forms, the encoding of real-world problems does usually not result in a clausal formula and although a possibility nonclausal formula is theoretically equivalent to some possibilistic clausal formula [26, 22], approaches needing clausal form transformations are practically infeasible or have experimentally shown to be highly inefficient as discussed below. Two kinds of clausal form transformation are known: (1) one is based on the repetitive application of the distributive laws to the input non-clausal formula until a logically equivalent clausal formula is obtained; and (2) the other transformation, Tsetin-transformation [59], is based on recursively substituting sub-formulas in the input non-clausal formula by fresh literals until obtaining an equi-satisfiable, but not equivalent, clausal formula.

The relevance of polynomial formula classes to deductive efficiency motivated their search, and currently, a great number of such classes is known. Nonetheless, they have been exclusively sought in the setting of clausal form and propositional logic, which is of course expressively limiting for real applications. As a consequence, a first polynomial propositional class in non-clausal (NC) form has recently been proposed. Along these lines and towards making NC tractability applicable beyond propositional logic, firstly, we define the Regular many-valued Horn Non-Clausal class, or RH, obtained by suitably amalgamating both regular classes: Horn and NC. Secondly, we demonstrate that the relationship between (1) RH and the regular Horn class is that syntactically RH subsumes the Horn class but that both classes are equivalent semantically; and between (2) RH and the regular non-clausal class is that RH contains all NC formulas whose clausal form is Horn. Thirdly, we define Regular Non-Clausal Unit-Resolution, or RUR-NC , and prove both that it is complete for RH and that checks its satisfiability in polynomial time. The latter fact shows that our intended goal is reached since RH is many-valued, non-clausal and tractable. As RH and RUR-NC are, both, basic in the DPLL scheme, the most efficient in propositional logic, and can be extended to some other non-classical logics, we argue that they pave the way for efficient non-clausal DPLL-based approximate reasoning.

Analogy-making is at the core of human intelligence and creativity with applications to such diverse tasks as commonsense reasoning, learning, language acquisition, and story telling. This paper contributes to the foundations of artificial general intelligence by introducing an abstract algebraic framework of analogical proportions of the form `$a$ is to $b$ what $c$ is to $d$' in the general setting of universal algebra. This enables us to compare mathematical objects possibly across different domains in a uniform way which is crucial for AI-systems. The main idea is to define solutions to analogical equations in terms of generalizations and to derive abstract terms of concrete elements from a `known' source domain which can then be instantiated in an `unknown' target domain to obtain analogous elements. We extensively compare our framework with two prominent and recently introduced frameworks of analogical proportions from the literature in the concrete domains of sets, numbers, and words and show that our framework yields strictly more reasonable solutions in all of these cases which provides evidence for the applicability of our framework. In a broader sense, this paper is a first step towards an algebraic theory of analogical reasoning and learning systems with potential applications to fundamental AI-problems like commonsense reasoning and computational learning and creativity.

The subject of this article is the introduction of a weaker concept of well-posedness of Bayesian inverse problems. The conventional concept of (`Lipschitz') well-posedness in [Stuart 2010, Acta Numerica 19, pp. 451-559] is difficult to verify in practice, especially when considering blackbox models, and probably too strong in many contexts. Our concept replaces the Lipschitz continuity of the posterior measure in the Hellinger distance by just continuity. This weakening is tolerable, since the continuity is in general only used as a stability criterion. The main result of this article is a proof of well-posedness for a large class of Bayesian inverse problems, where very little or no information about the underlying model is available. It includes any Bayesian inverse problem arising when observing finite-dimensional data perturbed by additive, non-degenerate Gaussian noise. Moreover, well-posedness with respect to other probability metrics is investigated, including weak convergence, total variation, Wasserstein, and also the Kullback-Leibler divergence.