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This paper considers a class of delay fractional optimal control problems with free terminal time. The fractional derivatives in this class of problems are described in the Caputo sense and they can be of different orders. We first show that for this class of problems, the well-known time-scaling transformation for mapping the free time horizon into a fixed time interval yields a new fractional-order system with variable time-delay. Then, we propose an explicit numerical scheme for solving the resulting fractional time-delay system, which gives rise to a discrete-time optimal control problem. Furthermore, we derive gradient formulas of the cost and constraint functions with respect to decision variables. On this basis, a gradient-based optimization approach is developed to solve the resulting discrete-time optimal control problem. Finally, an example problems is solved to demonstrate the effectiveness of our proposed solution approach. [ABSTRACT FROM AUTHOR]

In this paper, a computational method is proposed for solving a class of fractional optimal control problems subject to canonical constraints of equality and inequality. Fractional derivatives are described in the Atangana–Baleanu-Caputo sense, and their fractional orders can be different. To solve this problem, we present a discretization scheme based on the trapezoidal rule and a novel numerical integration technique. Then, the gradient formulas of the cost and constraint functions with respect to the decision variables are derived. Furthermore, a gradient-based optimization algorithm for solving the discretized optimal control problem is developed. Finally, the applicability and effectiveness of the proposed algorithm are verified through three non-trivial example problems. [ABSTRACT FROM AUTHOR]

This paper considers an optimal control problem governed by nonlinear fractional-order systems with multiple time-varying delays and subject to canonical constraints, where the fractional-order derivatives are expressed in the Caputo sense. To solve the problem by discretization scheme, an explicit numerical integration technique is proposed for solving the fractional-order system, and the trapezoidal rule is introduced to approximate the cost functional. Then, the gradients of the resulting cost and constraint functions are derived. On the basis of this result, we develop a gradient-based optimization algorithm to numerically solve the discretized problem. Finally, numerical results of several non-trivial examples are provided to illustrate the applicability and effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, a numerical method is developed for solving a class of delay fractional optimal control problems involving nonlinear time-delay systems and subject to state inequality constraints. The fractional derivatives in this class of problems are described in the sense of Caputo, and they can be of different orders. First, we propose a numerical integration scheme for the fractional time-delay system and prove that the convergence rate of the numerical solution to the exact one is of second order based on Taylor expansion and linear interpolation. This gives rise to a discrete-time optimal control problem. Then, we derive the gradient formulas of the cost and constraint functions with respect to the decision variables and present a gradient computation procedure. On this basis, a gradient-based optimization algorithm is developed to solve the resulting discrete-time optimal control problem. Finally, several example problems are solved to demonstrate the effectiveness of the developed solution approach. [ABSTRACT FROM AUTHOR]

Linear regression is one of the most important and widely used techniques in data analysis, for which a key step is the estimation of the unknown parameters. However, it is often carried out under the assumption that the full information of the error distribution is available. This is clearly unrealistic in practice. In this paper, we propose a distributionally robust formulation of L 1 -estimation (or the least absolute value estimation) problem, where the only knowledge on the error distribution is that it belongs to a well-defined ambiguity set. We then reformulate the estimation problem as a computationally tractable conic optimization problem by using duality theory. Finally, a numerical example is solved as a conic optimization problem to demonstrate the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose a bi-objective dynamic optimization model involving a nonlinear time-delay system to optimize the 1,3-propanediol (1,3-PD) production in a microbial batch process, where the productivity of 1,3-PD and the consumption rate of glycerol are taken as the two objectives. The initial concentrations of biomass and glycerol, and the terminal time of the process are the decision variables. By a time-scaling transformation, we first transform the problem to the one with fixed terminal time but involving a new system with variable time-delay. The normalized normal constraint method is then used to convert the resulting problem into a sequence of single-objective dynamic optimization problems. A gradient-based optimization method incorporating the constraint transcription technique is developed to solve each of these single-objective dynamic optimization problems. Finally, numerical results are provided to demonstrate the effectiveness of the proposed solution method. [ABSTRACT FROM AUTHOR]

Considering the hybrid nature in fed-batch culture of glycerol biconversion to 1,3-propanediol (1,3-PD) by Klebsiella pneumoniae, we propose a state-based switching dynamical system to describe the fermentation process. To maximize the concentration of 1,3-PD at the terminal time, an optimal switching control model subject to our proposed switching system and constraints of continuous state inequality and control function is presented. Because the number of the switchings is not known a priori, we reformulate the above optimal control problem as a two-level optimization problem. An optimization algorithm is developed to seek the optimal solution on the basis of a heuristic approach and control parametrization technique. Numerical results show that, by employing the obtained optimal control strategy, 1,3-PD concentration at the terminal time can be increased considerably. [ABSTRACT FROM AUTHOR]

Mathematical modelling of cellular metabolism plays an important role in understanding biological functions and providing identification of targets for biotechnological modification. This paper proposes a nonlinear bilevel programming (NBP) model to infer the objective function of anaerobic glycerol metabolism in Klebsiella Pneumoniae (K. Pneumoniae) for 1, 3-propanediol (1, 3-PD) production. Based on the Kuhn-Tucker optimality condition of the lower level problem, NBP is transformed into a nonlinear programming with complementary and slackness conditions. The authors give the existence theorem of solutions to NBP. An efficient algorithm is proposed to solve NBP and its convergence is also simply analyzed. Numerical results reveal some interesting conclusions, e.g., biomass production is the main force to drive glycerol metabolism, and the objective functions, which are obtained in term of several different groups of flux distributions, are similar. [ABSTRACT FROM AUTHOR]