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Semantics modulo satisfiability with applications: function representation, probabilities and game theory

Bulletin of Symbolic Logic 2022年第2期28卷 264-265页

作者： Sandro Márcio da Silva Preto Institute of Mathematics and Statistics University of São Paulo São Paulo Brazil 2021.

In the context of propositional logics, we apply semantics modulo satisfiability—a restricted semantics which comprehends only valuations that satisfy some specific set of formulas—with the aim to efficiently solve some computational tasks. Three possible such applications are *** begin by studying the possibility of implicitly representing rational McNaughton functions in Łukasiewicz Infinitely-valued Logic through semantics modulo satisfiability. We theoretically investigate some approaches to such representation concept, called representation modulo satisfiability, and describe a polynomial algorithm that builds representations in the newly introduced system. An implementation of the algorithm, test results and ways to randomly generate rational McNaughton functions for testing are presented. Moreover, we propose an application of such representations to the formal verification of properties of neural networks by means of the reasoning framework of Łukasiewicz Infinitely-valued ***, we move to the investigation of the satisfiability of joint probabilistic assignments to formulas of Łukasiewicz Infinitely-valued Logic, which is known to be an NP-complete problem. We provide an exact decision algorithm derived from the combination of linear algebraic methods with semantics modulo satisfiability. Also, we provide an implementation for such algorithm for which the phenomenon of phase transition is empirically ***, we study the game theory situation of observable games, which are games that are known to reach a Nash equilibrium, however, an external observer does not know what is the exact profile of actions that occur in a specific instance; thus, such observer assigns subjective probabilities to players actions. We study the decision problem of determining if a set of these probabilistic constraints is coherent by reducing it to the problems of satisfiability of probabilistic assignments to logical formulas both in Classical Propositional

关键词： Coherence of constraints formal methods function representation Łukasiewicz infinitely-valued logic Nash equilibrium neural networks non-classical probabilities piecewise linear functions probabilistic constraints probabilistic satisfiability propositional logics rational McNaughton functions uncertain games valuation semantics

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Cost efficiency measurement with price uncertainty: a data envelopment analysis

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MATHEMATICAL SCIENCES 2020年第4期14卷 387-396页

作者： Lotfi, F. Hosseinzadeh Amirteimoori, A. Moghaddas, Z. Vaez-Ghasemi, M. Islamic Azad Univ Dept Math Sci & Res Branch Tehran Iran Islamic Azad Univ Dept Math Qazvin Branch Qazvin Iran Islamic Azad Univ Dept Math Rasht Branch Rasht Iran

Data envelopment analysis (DEA) technique is commonly utilized for efficiency assessment in a variety of fields for both theoretical and applicational purposes. In classic cost efficiency measurement models, the input and output data and input prices should be known for each decision-making unit (DMU). However, in real-life markets the input prices are not precisely defined for DMUs. In this paper, we shed light on the fact that fixed prices assumption cannot reflect the reality of situations, because market will force lower prices if greater amounts of a product are purchased. It can be said that discounts are automatically considered in these circumstances. To this end, an innovative idea is considered to modify the classic cost efficiency DEA model in order to investigate the situations of real-life markets. Then, by an empirical example, a comparison between the proposed approach and the classic cost efficiency model is provided.

关键词： Data envelopment analysis Cost efficiency Mixed integer programming Marginal value piecewise linear functions

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SPICE Model of Current Polarity-Dependent piecewise linear Window Function for Memristors

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GAZI UNIVERSITY JOURNAL OF SCIENCE 2020年第4期33卷 766-777页

作者： Karakulak, Ertugrul Mutlu, Resat Tekirdag Namik Kemal Univ Vocat Sch Tech Sci Elect Dept Tekirdag Turkey Tekirdag Namik Kemal Univ Corlu Engn Fac Elect & Commun Engn Dept Corlu Tekirdag Turkey

Memristor and memristive systems are nonlinear systems. It is important to model them accurately. There are different memristor models and most of the models make use of window functions. In literature, there are various window functions. Recently, a piecewise linear (PWL) window function is used to model a memristor and memristive systems. Such a memristor with a PWL window function lacks a SPICE model. Also, in literature, there is current polarity dependent window functions proposed for memristors to model polarity dependent drift speed within the thin-film memristors. In this study, an alternative current-polarity dependent PWL window function is suggested to model a memristor, a different PWL function one for each current polarity is used, its SPICE model is made in LTSpice and also its simulation results are given. Such a model can be used to model the polarity dependent drift speed within the thin-film memristors.

关键词： Memristor piecewise linear functions Window functions Spice model

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Uniqueness of the Continuation of a Certain Function to a Positive Definite Function

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MATHEMATICAL NOTES 2020年第3-4期107卷 639-652页

作者： Manov, A. D. Donetsk Natl Univ UA-83114 Donetsk Ukraine

In 1940, M. G. Krein obtained necessary and sufficient conditions for the extension of a continuous function f defined in an interval (-a, a), a > 0, to a positive definite function on the whole number axis R. In addition, Krein showed that the function 1 - |x|, |x| < a, can be extended to a positive definite one on R if and only if 0 < a <= 2, and this function has a unique extension only in the case a = 2. The present paper deals with the problem of uniqueness of the extension of the function 1 - |x|, |x| <= a, a G (0,1), for a class of positive definite functions on R whose support is contained in the closed interval [-1,1] (the class T). It is proved that if a is an element of [1/2,1] and Re phi(x) = 1 - |x|, |x| <= a, for some phi is an element of T, then phi(x) = (1 - |x|) (+), x G R. In addition, for any a G (0,1/2), there exists a function phi is an element of T such that phi(x) = 1 - |x|, |x| <= a, but phi(x) not equal (1 - |x|)(+). Also the paper deals with extremal problems for positive definite functions and nonnegative trigonometric polynomials indirectly related to the extension problem under consideration.

关键词： extension of positive definite functions Bochner-Khinchine theorem piecewise linear functions nonnegative trigonometric polynomials extremal problems

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Weight normalized deep neural networks

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STAT 2021年第1期10卷

作者： Xu, Yixi Wang, Xiao Microsoft AI Good Res Lab Redmond WA USA Purdue Univ Dept Stat W Lafayette IN 47907 USA 150 N Univ St W Lafayette IN 47907 USA

The generalization error is the difference between the expected risk and the empirical risk of a learning algorithm. This generalization error can be upper bounded by the Rademacher complexity of the underlying hypothesis class with high probability. This paper studies the function class of L-p,L- q weight normalized deep neural networks. We present a general framework for norm-based capacity control and functional characterization for this class. In particular, we have established the upper bound on the Rademacher complexities of this family. Especially, with an L-1,L- infinity normalization, we discuss properties of a width-independent capacity control, which only depends on the depth by a square root term. Furthermore, if the activation functions are anti-symmetric, the bound on the Rademacher complexity is independent of both the width and the depth up to a log factor.

关键词： deep neural networks generalization error piecewise linear functions Rademacher complexity sparsity weight normalization

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The impacts of convex piecewise linear cost formulations on AC optimal power flow

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ELECTRIC POWER SYSTEMS RESEARCH 2021年 199卷

作者： Coffrin, Carleton Knueven, Bernard Holzer, Jesse Vuffray, Marc Los Alamos Natl Lab Los Alamos NM 87545 USA Natl Renewable Energy Lab Golden CO 80401 USA Pacific Northwest Natl Lab Richland WA 99354 USA

Despite strong connections through shared application areas, research efforts on power market optimization (e. g., unit commitment) and power network optimization (e.g., optimal power flow) remain largely independent. A notable illustration of this is the treatment of power generation cost functions, where nonlinear network optimization has largely used polynomial representations and market optimization has adopted piecewise linear encodings. This work combines state-of-the-art results from both lines of research to understand the best mathematical formulations of the nonlinear AC optimal power flow problem with piecewise linear generation cost functions. An extensive numerical analysis of non-convex models, linear approximations, and convex relaxations across fifty-four realistic test cases illustrates that nonlinear optimization methods are surprisingly sensitive to the mathematical formulation of piecewise linear functions. The results indicate that a poor formulation choice can slow down algorithm performance by a factor of ten, increasing the runtime from seconds to minutes. These results provide valuable insights into the best formulations of nonlinear optimal power flow problems with piecewise linear cost functions, an important step towards building a new generation of energy markets that incorporate the nonlinear AC power flow model.

关键词： Nonlinear optimization Convex optimization Optimal power flow piecewise linear functions

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Study of the numerical efficiency of structured abs-normal forms

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OPTIMIZATION METHODS & SOFTWARE 2019年第5期36卷 909-933页

作者： Bosse, Torsten Falko Narayanan, Sri Hari Krishna Friedrich Schiller Univ Jena Adv Comp Jena Germany Argonne Natl Lab MCS Lemont IL USA

The abs-normal form (ANF) can be used to represent almost any piecewise linear function. Several of these piecewise linear functions exhibit a certain structure that has an impact on the numerical efficiency of the ANF representation. In this paper, three common structures are investigated that typically arise in applications and require special numerical treatment: the sum, the composition, and the component-wise maximum/minimum of several functions. For these structures, the corresponding expressions of the resulting abs-normal form are provided, as well as some alternatives. The theoretical observations are supported by numerical results.

关键词： piecewise linear functions algorithmic Differentiation abs Normal Form structural Investigation

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A note on surjectivity of piecewise affine mappings

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OPTIMIZATION LETTERS 2019年第2期13卷 439-443页

作者： Radons, Manuel Tech Univ Berlin Workgrp Discrete Math Geometry Berlin Germany

A standard theorem in nonsmooth analysis states that a piecewise affine function F:RnRn is surjective if it is coherently oriented in that the linear parts of its selection functions all have the same nonzero determinant sign. In this note we prove that surjectivity already follows from coherent orientation of the selection functions which are active on the unbounded sets of a polyhedral subdivision of the domain corresponding to F. A side bonus of the argumentation is a short proof of the classical statement that an injective piecewise affine function is coherently oriented.

关键词： Nonsmooth optimization piecewise affine functions piecewise linear functions Surjectivity criteria

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Saddle representations of positively homogeneous functions by linear functions

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OPTIMIZATION LETTERS 2018年第8期12卷 1971-1980页

作者： Gorokhovik, Valentin V. Trafimovich, Marina Natl Acad Sci Belarus Inst Math Ul Surganova 11 Minsk 220072 BELARUS Sukhoi State Tech Univ Gomel Pr Octiabria 48 Gomel 246746 BELARUS

We say that a positively homogeneous function admits a saddle representation by linear functions iff it admits both an inf-sup-representation and a sup-inf-representation with the same two-index family of linear functions. In the paper we show that each continuous positively homogeneous function can be associated with a two-index family of linear functions which provides its saddle representation. We also establish characteristic properties of those two-index families of linear functions which provides saddle representations of functions belonging to the subspace of Lipschitz continuous positively homogeneous functions as well as the subspaces of difference sublinear and piecewise linear functions.

关键词： Positively homogeneous functions Saddle representation Lipschitz continuity Difference sublinear functions piecewise linear functions

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Critical multipliers in variational systems via second-order generalized differentiation

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MATHEMATICAL PROGRAMMING 2018年第2期169卷 605-648页

作者： Mordukhovich, Boris S. Sarabi, M. Ebrahim Wayne State Univ Dept Math Detroit MI 48202 USA RUDN Univ Moscow 117198 Russia Miami Univ Dept Math Oxford OH 45065 USA

In this paper we introduce the notions of critical and noncritical multipliers for variational systems and extend to a general framework the corresponding notions by Izmailov and Solodov developed for classical Karush-Kuhn-Tucker (KKT) systems. It has been well recognized that critical multipliers are largely responsible for slow convergence of major primal-dual algorithms of optimization. The approach of this paper allows us to cover KKT systems arising in various classes of smooth and nonsmooth problems of constrained optimization including composite optimization, minimax problems, etc. Concentrating on a polyhedral subdifferential case and employing recent results of second-order subdifferential theory, we obtain complete characterizations of critical and noncritical multipliers via the problem data. It is shown that noncriticality is equivalent to a certain calmness property of a perturbed variational system and that critical multipliers can be ruled out by full stability of local minimizers in problems of composite optimization. For the latter class we establish the equivalence between noncriticality of multipliers and robust isolated calmness of the associated solution map and then derive explicit characterizations of these notions via appropriate second-order sufficient conditions. It is finally proved that the Lipschitz-like/Aubin property of solution maps yields their robust isolated calmness.

关键词： Variational systems Composite optimization Critical and noncritical multipliers Generalized differentiation piecewise linear functions Robust isolated calmness Lipschitzian stability

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