Ozlem Defterli

A Central Difference Numerical Scheme for Fractional Optimal Control Problems

This paper presents a modified numerical scheme for a class of fractional optimal control problems where a fractional derivative (FD) is defined in the Riemann—Liouville sense. In this scheme, the entire time domain is divided into several sub-domains, and a FD at a time node point is approximated using a modified Grünwald—Letnikov approach. For the first-order derivative, the proposed modified Grünwald— Letnikov definition leads to a central difference scheme. When the approximations are substituted into the fractional optimal control equations, it leads to a set of algebraic equations which are solved using a direct numerical technique. Two examples, one time-invariant and the other time-variant, are considered to study the performance of the numerical scheme. Results show that 1) as the order of the derivative approaches an integer value, these formulations lead to solutions for the integer-order system, and 2) as the sizes of the sub-domains are reduced, the solutions converge. It is hoped that the present scheme would lead to stable numerical methods for fractional differential equations and optimal control problems.

Fractional Optimal Control Problems with Several State and Control Variables

In many applications, fractional derivatives provide better descriptions of the behavior of dynamic systems than other techniques. For this reason, fractional calculus has been used to analyze systems having noninteger order dynamics and to solve fractional optimal control problems. In this study, we describe a formulation for fractional optimal control problems defined in multi-dimensions. We consider the case where the dimensions of the state and control variables are different from each other. Riemann—Liouville fractional derivatives are used to formulate the problem. The fractional differential equations involving the state and control variables are solved using Grünwald—Letnikov approximation. The performance of the formulation is shown using an example.

Modeling, inference and optimization of regulatory networks based on time series data

In this survey paper, we present advances achieved during the last years in the development and use of OR, in particular, optimization methods in the new gene-environment and eco-finance networks, based on usually finite data series, with an emphasis on uncertainty in them and in the interactions of the model items. Indeed, our networks represent models in the form of time-continuous and time-discrete dynamics, whose unknown parameters we estimate under constraints on complexity and regularization by various kinds of optimization techniques, ranging from linear, mixed-integer, spline, semi-infinite and robust optimization to conic, e.g., semi-definite programming. We present different kinds of uncertainties and a new time-discretization technique, address aspects of data preprocessing and of stability, related aspects from game theory and financial mathematics, we work out structural frontiers and discuss chances for future research and OR application in our real world.

A numerical scheme for two-dimensional optimal control problems with memory effect

A new formulation for multi-dimensional fractional optimal control problems is presented in this article. The fractional derivatives which are coming from the formulation of the problem are defined in the Riemann-Liouville sense. Some terminal conditions are imposed on the state and control variables whose dimensions need not be the same. A numerical scheme is described by using the Grunwald-Letnikov definition to approximate the Riemann-Liouville Fractional Derivatives. The set of fractional differential equations, which are obtained after the discretization of the time domain, are solved within the Grunwald-Letnikov approximation to obtain the state and the control variable numerically. A two-dimensional fractional optimal control problem is studied as an example to demonstrate the performance of the scheme.

The new robust conic GPLM method with an application to finance: prediction of credit default

This paper contributes to classification and identification in modern finance through advanced optimization. In the last few decades, financial misalignments and, thereby, financial crises have been increasing in numbers due to the rearrangement of the financial world. In this study, as one of the most remarkable of these, countries’ debt crises, which result from illiquidity, are tried to predict with some macroeconomic variables. The methodology consists of a combination of two predictive regression models, logistic regression and robust conic multivariate adaptive regression splines (RCMARS), as linear and nonlinear parts of a generalized partial linear model. RCMARS has an advantage of coping with the noise in both input and output data and of obtaining more consistent optimization results than CMARS. An advanced version of conic generalized partial linear model which includes robustification of the data set is introduced: robust conic generalized partial linear model (RCGPLM). This new model is applied on a data set that belongs to 45 emerging markets with 1,019 observations between the years 1980 and 2005.

Advanced Mathematical and Statistical Tools in the Dynamic Modeling and Simulation of Gene-Environment Regulatory Networks

In this study, some methodologies and a review of the recently obtained new results are presented for the problem of modeling, anticipation and forecasting of genetic regulatory systems, as complex systems. In this respect, such kind of complex systems are modeled in the dynamical sense into the two different ways, namely, by a system of ordinary differential equations (ODEs) and Gaussian graphical methods (GGM). An artificial time-course microarray dataset of a gene-network is modeled as an example by using both ODE method and GGM. In this analysis, since the actual interactions of the nodes, i.e., genes, are assumed to be unknown, the discrete time measurements are initially used for the inference of the system’s interactions, i.e., the edges between nodes, by the underlying two methods. Then, the results of inference from ordinary differential equation based model are applied to a class of previously developed new numerical schemes for the generation of further states of the system. In this simulation, we present the recent results of a set of explicit Runge-Kutta methods that are implemented.

Killing‐Yano tensors, surface terms and superintegrable systems

Killing‐Yano and Killing tensors are investigated corresponding to a set of two dimensional superintegrable systems. A suitable surface term is added to the corresponding free Lagrangian describing the motion of a particle on a 2‐sphere of unit radius and we analyze the symmetries of the obtained geometries.

Fuzzy prediction strategies for gene-environment networks – Fuzzy regression analysis for two-modal regulatory systems

Target-environment networks provide a conceptual framework for the analysis and prediction of complex regulatory systems such as genetic networks, eco-finance networks or sensor-target assignments. These evolving networks consist of two major groups of entities that are interacting by unknown relationships. The structure and dynamics of the hidden regulatory system have to be revealed from uncertain measurement data. In this paper, the concept of fuzzy target-environment networks is introduced and various fuzzy possibilistic regression models are presented. The relation between the targets and/or environmental entities of the regulatory network is given in terms of a fuzzy model. The vagueness of the regulatory system results from the (unknown) fuzzy coefficients. For an identification of the fuzzy coefficients’ shape, methods from fuzzy regression are adapted and made applicable to the bi-level situation of target-environment networks and uncertain data. Various shapes of fuzzy coefficients are considered and the control of outliers is discussed. A first numerical example is presented for purposes of illustration. The paper ends with a conclusion and an outlook to future studies.

Vester's Sensitivity Model for Genetic Networks with Time-Discrete Dynamics

We propose a new method to explore the characteristics of genetic networks whose dynamics are described by a linear discrete dynamical model x t + 1 = Ax t . The gene expression data x t is given for various time points and the matrix A of interactions among the genes is unknown. First we formulate and solve a parameter estimation problem by linear programming in order to obtain the entries of the matrix A. We then use ideas from Vester’s Sensitivity Model, more precisely, the Impact Matrix, and the determination of the Systemic Roles, to understand the interactions among the genes and their role in the system. The method identifies prominent outliers, that is, the most active, reactive, buffering and critical genes in the network. Numerical examples for different datasets containing mRNA transcript levels during the cell cycle of budding yeast are presented.

Corrigendum: Corrigendum to A numerical scheme for two dimensional optimal control problems with memory effect [Comput. Math. Appl. 59 (2010) 1630-1636]

The author presents in below the necessary corrections on page 1631 of the original article ‘‘A numerical scheme for two-dimensional optimal control problems with memory effect’’ [Comput. Math. Appl. 59 (2010) 1630-1636]: – The text immediately after Equation 4 should read: ‘‘and satisfying the terminal condition x(a) = c. Here t denotes the time, x(t) and u(t) are a nx × 1 state and nu × 1 control vectors (not necessarily in same dimension), f and g are a scalar and a nx× 1 vector functions, aDαt x is the left RLFD of order α of xwith respect to t , and c is a given vector.’’ instead of ‘‘and satisfying the terminal conditions x(a) = c and x(b) = d. Here t denotes the time, x(t) and u(t) are a nx × 1 state and nu× 1 control vectors (not necessarily in same dimension), f and g are a scalar and a nx× 1 vector functions, aDαt x is the left RLFD of order α of xwith respect to t , and c , d are given vectors.’’ – The text of the first sentence of the paragraph after Equation 10 should read: ‘‘The equations in (7)–(9) describe the necessary conditions in terms of a Hamiltonian for the FOCP defined above.’’ instead of ‘‘The equations in (7)–(10) describe the necessary conditions in terms of a Hamiltonian for the FOCP defined above.’’