Fuzzy Logic

Fuzzy Logic is becoming an essential method of solving problems in all domains. It gives tremendous impact on the design of autonomous intelligent systems. The purpose of this book is to introduce Hybrid Algorithms, Techniques, and Implementations of Fuzzy Logic. The book consists of thirteen chapters highlighting models and principles of fuzzy logic and issues on its techniques and implementations. The intended readers of this book are engineers, researchers, and graduate students interested in fuzzy logic systems

Supervised classification and mathematical optimization

Data Mining techniques often ask for the resolution of optimization problems. Supervised Classification, and, in particular, Support Vector Machines, can be seen as a paradigmatic instance. In this paper, some links between Mathematical Optimization methods and Supervised Classification are emphasized. It is shown that many different areas of Mathematical Optimization play a central role in off-the-shelf Supervised Classification methods. Moreover, Mathematical Optimization turns out to be extremely useful to address important issues in Classification, such as identifying relevant variables, improving the interpretability of classifiers or dealing with vagueness/noise in the data.Ministerio de Ciencia e InnovaciónJunta de Andalucí

A systemic framework for the computational analysis of complex economies: An evolutionary-institutional perspective on the ontology, epistemology, and methodology of complexity economics

This theses introduces the idea of a symbiotic relationship between evolutionary-institutional and complexity economics. It consists of two major contributions: The first contribution focuses on how the emerging research program of complexity economics can benefit from evolutionary-institutional theory. I show that complexity economics still lacks an adequate philosophical foundation. I explicate why such a foundation is needed if complexity economics is to promote further scientific progress and that such a foundation must consist of an adequate ontology, epistemology, and methodology. The following parts of the theses then draw upon institutionalist and social theory to develop these three aspects: I derive a definition of complex economic systems by identifying their essential properties. I then propose an epistemology that is based on the concepts of mechanism-based explanation, generative sufficiency, and an extended version of Uskali Mäki's concept of â Models as Isolations and Surrogate Systemsâ . I continue with some methodological considerations and argue that the method of 'Agent based computational economic modeling' must play a distinctive role for the analysis of complex economies. The second contribution of the theses shows how evolutionary-institutionalism can profit from a methodological transfer from complexity economics. In particular I argue that the method of 'Agent based computational modeling' can advance institutionalism both as a formalization device and by providing theoretical concepts that are useful for institutionalist theorizing itself. The theses closes by discussing a potential convergence of evolutionary-institutional and complexity economics and gives an outlook on avenues for further research

A Primal-Dual Augmented Lagrangian Penalty-Interior-Point Algorithm for Nonlinear Programming

This thesis treats a new numerical solution method for large-scale nonlinear optimization problems. Nonlinear programs occur in a wide range of engineering and academic applications like discretized optimal control processes and parameter identification of physical systems. The most efficient and robust solution approaches for this problem class have been shown to be sequential quadratic programming and primal-dual interior-point methods. The proposed algorithm combines a variant of the latter with a special penalty function to increase its robustness due to an automatic regularization of the nonlinear constraints caused by the penalty term. In detail, a modified barrier function and a primal-dual augmented Lagrangian approach with an exact l2-penalty is used. Both share the property that for certain Lagrangian multiplier estimates the barrier and penalty parameter do not have to converge to zero or diverge, respectively. This improves the conditioning of the internal linear equation systems near the optimal solution, handles rank-deficiency of the constraint derivatives for all non-feasible iterates and helps with identifying infeasible problem formulations. Although the resulting merit function is non-smooth, a certain step direction is a guaranteed descent. The algorithm includes an adaptive update strategy for the barrier and penalty parameters as well as the Lagrangian multiplier estimates based on a sensitivity analysis. Global convergence is proven to yield a first-order optimal solution, a certificate of infeasibility or a Fritz-John point and is maintained by combining the merit function with a filter or piecewise linear penalty function. Unlike the majority of filter methods, no separate feasibility restoration phase is required. For a fixed barrier parameter the method has a quadratic order of convergence. Furthermore, a sensitivity based iterative refinement strategy is developed to approximate the optimal solution of a parameter dependent nonlinear program under parameter changes. It exploits special sensitivity derivative approximations and converges locally with a linear convergence order to a feasible point that further satisfies the perturbed complementarity condition of the modified barrier method. Thereby, active-set changes from active to inactive can be handled. Due to a certain update of the Lagrangian multiplier estimate, the refinement is suitable in the context of warmstarting the penalty-interior-point approach. A special focus of the thesis is the development of an algorithm with excellent performance in practice. Details on an implementation of the proposed primal-dual penalty-interior-point algorithm in the nonlinear programming solver WORHP and a numerical study based on the CUTEst test collection is provided. The efficiency and robustness of the algorithm is further compared to state-of-the-art nonlinear programming solvers, in particular the interior-point solvers IPOPT and KNITRO as well as the sequential quadratic programming solvers SNOPT and WORHP

Memory, multiple equilibria and emerging market crises

We present a new Generalized Markov Equilibrium (GME) approach to studying sudden stops and financial crises in emerging countries in the canonical small open economy model with equilibrium price-dependent collateral constraints. Our approach to characterizing and computing stochastic equilibrium dynamics is global, encompasses recursive equilibrium as a special case, yet allows for a much more flexible approach to modeling memory in such models that are known to have multiple equilibrium. We prove the existence of ergodic GME selections from the set of sequential competitive equilibrium, and show that at the same time ergodic GME selectors can replicate all the observed phases of the macro crises associated with a sudden stop (boom, collapse, spiralized recession, recovery) while still being able to capture the long-run stylized behavior of the data. We also compute stochastic equilibrium dynamics associated with stationary and nonstationary GME selections, and we find that a) the ergodic GME selectors generate stochastic dynamics that are less financially constrained with respect to stationary non-ergodic paths, b) non-stationary GME selections exhibit a great range of fluctuations in macroeconomic aggregates compared to the stationary selections. From a theoretical perspective, we prove the existence of both sequential competitive equilibrium and (minimal state space) recursive equilibrium, as well as provide a complete theory of robust recursive equilibrium comparative statics in deep parameters. Consistent with recent results in the literature, relative to the set of recursive equilibrium, we find 2 stationary equilibrium: one with high/over borrowing, the other with low/under borrowing. These equilibrium are extremal and “selffulfilling” under rational expectations. The selection among these equilibria depend on observable variables and not on sunspots

Toll competition in highway transportation networks

Within a highway transportation network, the social welfare implications of two different groups of agents setting tolls in competition for revenues are studied. The first group comprises private sector toll road operators aiming to maximise revenues. The second group comprises local governments or jurisdictions who may engage in tax exporting. Extending insights from the public economics literature, jurisdictions tax export because when setting tolls to maximise welfare for their electorate, they simultaneously benefit from revenues from extra-jurisdictional users. Hence the tolls levied by both groups will be higher than those intended solely to internalise congestion, which then results in welfare losses. Therefore the overarching question investigated is the extent of welfare losses stemming from such competition for toll revenues. While these groups of agents are separately studied, the interactions between agents in each group in competition can be modelled within the common framework of Equilibrium Problems with Equilibrium Constraints. Several solution algorithms, adapting methodologies from microeconomics as well as evolutionary computation, are proposed to identify Nash Equilibrium toll levels. These are demonstrated on realistic transportation networks. As an alternative paradigm to competition, the possibilities for co-operation between agents in each group are also explored. In the case of toll road operators, the welfare consequences of competition could be positive or adverse depending on the interrelationships between the toll roads in competition. The results therefore generalise those previously obtained to a more realistic setting investigated here. In the case of competition between jurisdictions, it is shown that the fiscal externality of tax exporting resulting from their toll setting decisions can substantially reduce the welfare gains from internalising congestion. The ability of regulation, co-operation and bilateral bargaining to reduce the welfare losses are assessed. The research thus contributes to informing debates regarding the appropriate level of institutional governance for toll pricing policies

Preconditioned iterative methods for optimal control problems with time-dependent PDEs as constraints

In this work, we study fast and robust solvers for optimal control problems with Partial Differential Equations (PDEs) as constraints. Speci cally, we devise preconditioned iterative methods for time-dependent PDE-constrained optimization problems, usually when a higher-order discretization method in time is employed as opposed to most previous solvers. We also consider the control of stationary problems arising in uid dynamics, as well as that of unsteady Fractional Differential Equations (FDEs). The preconditioners we derive are employed within an appropriate Krylov subspace method. The fi rst key contribution of this thesis involves the study of fast and robust preconditioned iterative solution strategies for the all-at-once solution of optimal control problems with time-dependent PDEs as constraints, when a higher-order discretization method in time is employed. In fact, as opposed to most work in preconditioning this class of problems, where a ( first-order accurate) backward Euler method is used for the discretization of the time derivative, we employ a (second-order accurate) Crank-Nicolson method in time. By applying a carefully tailored invertible transformation, we symmetrize the system obtained, and then derive a preconditioner for the resulting matrix. We prove optimality of the preconditioner through bounds on the eigenvalues, and test our solver against a widely-used preconditioner for the linear system arising from a backward Euler discretization. These theoretical and numerical results demonstrate the effectiveness and robustness of our solver with respect to mesh-sizes and regularization parameter. Then, the optimal preconditioner so derived is generalized from the heat control problem to time-dependent convection{diffusion control with Crank- Nicolson discretization in time. Again, we prove optimality of the approximations of the main blocks of the preconditioner through bounds on the eigenvalues, and, through a range of numerical experiments, show the effectiveness and robustness of our approach with respect to all the parameters involved in the problem. For the next substantial contribution of this work, we focus our attention on the control of problems arising in fluid dynamics, speci fically, the Stokes and the Navier-Stokes equations. We fi rstly derive fast and effective preconditioned iterative methods for the stationary and time-dependent Stokes control problems, then generalize those methods to the case of the corresponding Navier-Stokes control problems when employing an Oseen approximation to the non-linear term. The key ingredients of the solvers are a saddle-point type approximation for the linear systems, an inner iteration for the (1,1)-block accelerated by a preconditioner for convection-diffusion control problems, and an approximation to the Schur complement based on a potent commutator argument applied to an appropriate block matrix. Through a range of numerical experiments, we show the effectiveness of our approximations, and observe their considerable parameter-robustness. The fi nal chapter of this work is devoted to the derivation of efficient and robust solvers for convex quadratic FDE-constrained optimization problems, with box constraints on the state and/or control variables. By employing an Alternating Direction Method of Multipliers for solving the non-linear problem, one can separate the equality from the inequality constraints, solving the equality constraints and then updating the current approximation of the solutions. In order to solve the equality constraints, a preconditioner based on multilevel circulant matrices is derived, and then employed within an appropriate preconditioned Krylov subspace method. Numerical results show the e ciency and scalability of the strategy, with the cost of the overall process being proportional to N log N, where N is the dimension of the problem under examination. Moreover, the strategy presented allows the storage of a highly dense system, due to the memory required being proportional to N

Memoria, equilibrios múltiples y crisis en países emergentes

We present a new Generalized Markov Equilibrium (GME) approach to studying sudden stops and financial crises in emerging countries in the canonical small open economy model with equilib-rium price-dependent collateral constraints. Our approach to characterizing and computing stochastic equilibrium dynamics is global, encompasses recursive equilibrium as a special case, yet allows for a much more flexible approach to modeling memory in such models that are known to have multiple equilibrium. We prove the existence of ergodic GME selections from the set of sequential competitive equilibrium, and show that at the same time ergodic GME selectors can replicate all the observed phases of the macro crises associated with a sudden stop (boom, collapse, spiralized recession, recov-ery) while still being able to capture the long-run stylized behavior of the data. We also compute stochastic equilibrium dynamics associated with stationary and nonstationary GME selections, and we find that: a) the ergodic GME selectors generate stochastic dynamics that are less financially constrained with respect to stationary non-ergodic paths; and, b) non-stationary GME selections ex-hibit a great range of fluctuations in macroeconomic aggregates compared to the stationary selections. From a theoretical perspective, we prove the existence of both sequential competitive equilibrium and (minimal state space) recursive equilibrium, as well as provide a complete theory of robust recursive equilibrium comparative statics in deep parameters. Consistent with recent results in the literature, relative to the set of recursive equilibrium, we find 2 stationary equilibrium: one with high/over borrowing, the other with low/under borrowing. These equilibrium are extremal and “self-fulfilling” under rational expectations. The selection among these equilibria depend on observable variables and not on sunspots.Presentamos un nuevo equilibrio generalizado de Markov (GME) para estudiar crisis de balanza de pagos en economías emergentes que sufren fricciones financieras en la forma de restricciones de colateral. Nuestro enfoque permite caracterizar y computar los equilibrios dinámicos y estocásticos en forma global, comprende a otros equilibrios recursivos (como los de espacio de esta mínimo) y representa una forma flexible de modelar “memoria” en estas economías que suelen tener equilibrios múltiples. Probamos la existencia de un GME ergódico cómo una selección del equilibrio secuencial el cual a su vez puede replicar tanto todas las fases de una crisis de balanza de pagos cómo los hechos estilizados de largo plazo. Computamos el equilibrio ergódico, estacionario y no estacionario. Encontramos que el equilibrio ergódico tiene trayectorias del consumo más suaves y que el no estacionario puede replicar un gran rango de crisis de balanza de pagos. Desde una perspectiva teórica, probamos la existencia del equilibrio secuencial y recursivo en espacio de estados mínimos, como así también resultados de estática comparativa robusta. En línea con la literatura, encontramos 2 tipos de equilibrios estacionarios: uno con alto y el otro con bajo endeudamiento. Estos equilibrios son auto-validantes en expectativas racionales y no requieren de manchas solares para su coordinación

New Development of Neutrosophic Probability, Neutrosophic Statistics, Neutrosophic Algebraic Structures, and Neutrosophic & Plithogenic Optimizations

This Special Issue puts forward for discussion state-of-the-art papers on new topics related to neutrosophic theories, such as neutrosophic algebraic structures, neutrosophic triplet algebraic structures, neutrosophic extended triplet algebraic structures, neutrosophic algebraic hyperstructures, neutrosophic triplet algebraic hyperstructures, neutrosophic n-ary algebraic structures, neutrosophic n-ary algebraic hyperstructures, refined neutrosophic algebraic structures, refined neutrosophic algebraic hyperstructures, quadruple neutrosophic algebraic structures, refined quadruple neutrosophic algebraic structures, neutrosophic image processing, neutrosophic image classification, neutrosophic computer vision, neutrosophic machine learning, neutrosophic artificial intelligence, neutrosophic data analytics, neutrosophic deep learning, neutrosophic symmetry, and their applications in the real world. This book leads to the further advancement of the neutrosophic and plithogenic theories of NeutroAlgebra and AntiAlgebra, NeutroGeometry and AntiGeometry, Neutrosophic n-SuperHyperGraph (the most general form of graph of today), Neutrosophic Statistics, Plithogenic Logic as a generalization of MultiVariate Logic, Plithogenic Probability and Plithogenic Statistics as a generalization of MultiVariate Probability and Statistics, respectively, and presents their countless applications in our every-day world