On efficiency and mixed duality for a new class of nonconvex multiobjective variational control problems

In the paper, we introduce the concepts of $$G$$ -type I and generalized $$G$$ -type I functions for a new class of nonconvex multiobjective variational control problems. For such nonconvex vector optimization problems, we prove sufficient optimality conditions for weakly efficiency, efficiency and properly efficiency under assumptions that the functions constituting them are $$G$$ -type I and/or generalized $$G$$ -type I objective and constraint functions. Further, for the considered multiobjective variational control problem, its dual multiobjective variational control problem is given and several duality results are established under (generalized) $$G$$ -type I objective and constraint functions.

In this paper, by using the new concept of (ϱ,ψ,ω)-quasiinvexity associated with interval-valued path-independent curvilinear integral functionals, we establish some duality results for a new class of multiobjective variational control problems with interval-valued components. More concretely, we formulate and prove weak, strong, and converse duality theorems under (ϱ,ψ,ω)-quasiinvexity hypotheses for the considered class of optimization problems.

Our main goal in this chapter is to discuss the application of Alternating Direction Methods of Multipliers (ADMM) to the numerical solution of non-convex (and possibly non-smooth) variational problems. After giving a relatively detailed history of the ADMM methodology, we will discuss its application to the solution of problems from nonlinear Continuum Mechanics, nonlinear Elasticity, in particular. The ADMM solution of the two-dimensional Dirichlet problem for the Monge-Ampere equation will be discussed also. The results of numerical experiments will be reported, in order to illustrate the capabilities of the methodology under consideration

This paper investigates a class of multi-objective variational control problems, named (VCP), characterized by vector-valued integral objective/cost functional and differentiable inequality constraints. More precisely, we formulate the problem in a general framework and, in order to analyze optimality criteria, we associate scalar subproblems with the multi-objective setting and establish conditions under which efficient solutions of the original vector problem correspond to optimal solutions of these scalarized problems. We derive the necessary optimality conditions in the form of Euler–Lagrange-type equations, involving multipliers and piecewise smooth functions as variables. Furthermore, we introduce the new notion of (ϵ0, ϵ1) − (c0, c1) − type − I pair of functionals, adapted to the variational control setting. The introduced generalized convexity concepts are very important in characterizing efficient solutions and in deriving enhanced optimality conditions. These results enrich the theoretical framework of multi-objective variational control and provide a solid basis for future research and applications in mathematical optimization and control theory.

In this paper, we develop a multi-objective model to optimally control the lead time of a manufacturing system, using goal attainment method. The manufacturing system is modelled as a new queueing system based on the failure rate of products, in which the products may have quality problems and need to be repaired in each service station. It is assumed that the product order arrives according to a Poisson process. In each service station, there is only one server (machines) with exponentially distributed processing time, in which the service rate (capacity) is controllable. The maintenance time of manufacturing and assembly operations is exponentially distributed. By using the total probability decomposition and Markov process method, the Queue Length Distribution, the recursion expression of the equilibrium distribution and the mean waiting time are obtained. Then, we develop a multi-objective optimal control problem, in which the average lead time, the maintenance costs and the operating costs of the system per period are minimized using goal attainment method.

Nonconvex variational problems with anisotropic perturbations

A PID-type controller incorporating neural network elimination scheme and sliding-mode control action for different objectives including H/sub 2/ tracking performance, H/sub /spl infin// tracking performance, and regional pole constraints is developed in robotic systems under plant uncertainties and external disturbances. The adaptive neural networks are used to compensate the plant uncertainties. The sliding-mode control action is included to eliminate the effect of approximation error via neural network approximation. The sufficient conditions are developed for different objectives in terms of linear matrix inequality (LMI) formulations. The interesting combinations of different objectives are considered in this paper, which include H/sub /spl infin// PID tracking control design with regional pole constraints and mixed H/sub 2//H/sub /spl infin// PID tracking control design with regional pole constraints. These multiobjective PID control problems are characterized in terms of eigenvalue problem (EVP). The EVP can be efficiently solved by the LMI toolbox in Matlab. The proposed methods are simple and the PID control gain for different objectives can be obtained systematically.

In this work, we introduce a pair of multiobjective second-order symmetric dual variational problems. Weak, strong, and converse duality theorems for this pair are established under the assumption of ?-bonvexity/?-pseudobonvexity. At the end, the static case of our problems has also been discussed.

This paper studies multi-objective control of a full-vehicle suspension excited by random road disturbances. The control problem is first formulated as a mixed ℋ2/ℋ∞ synthesis problem and an output-feedback solution is obtained by using linear-matrix-inequalities. Next, the multi-objective control problem is re-formulated as a non-convex and non-smooth optimisation problem with controller order restricted to be less than the vehicle model order. For a range of orders, controllers are synthesised by using the HIFOO toolbox. The efficacy of the presented procedures are demonstrated by several design examples.

To comply with the effluent standards and growing demands for safety and reliability, the operation of wastewater treatment processes (WWTPs) has been considered as a multiobjective control problem. In this article, a data-driven multiobjective predictive control (MOPC) method is developed to deal with the conflicting control objectives to improve the operation performance of WWTPs. The main contributions of MOPC are three folds: first, a multiobjective control strategy is developed in the design of MOPC. And an adaptive fuzzy neural network identifier, using the relevant process data, is designed to catch the nonlinear behaviors of WWTPs. Second, a transfer multiobjective optimization algorithm (TMOOA) is developed to obtain the optimal solutions of the conflicting control objectives. The major advantage of TMOOA is its low computational cost, which is realized by avoiding the computation of Pareto fronts. Third, the stability of MOPC has been given in detail. Meanwhile, the benefits and feasibility of MOPC are confirmed on the benchmark simulation model no. 2. The results further demonstrate the effectiveness of the proposed control method.

Numerical approximations of strongly degenerate variational problems of the form J ( u ) = ∫ 0 1 F ( u ′ ) + ( u − f ) 2 J(u) = \smallint _0^1F(u’ ) + {(u - f)^2} are considered, where F is assumed convex but may have intervals where F = 0 F = 0 . It is shown that, in spite of the degeneracy, natural numerical approximations still converge in W 1 , p {W^{1,p}} . Rates in weaker norms and the connection with nonconvex variational problems are also considered.

Multiobjective Control Problem with Generalized Invexity

This paper proposes a multi-objective predictive control based on fuzzy satisfactory goal programming. For the different control goals, the multi-objective predictive control problem with priority is formulated. In order to acquire the preferred result, the fuzzy satisfactory goal programming for three types of fuzzy relations is presented. The released order of the satisfactory degrees is modelled to replace priority according to that the higher priority achieves the higher satisfactory degree. The balance between optimization and priority is realized by regulating the parameter. The example shows the proposed method's effectiveness.

A multi-objective disturbance attenuation problem is considered as a novel framework for control and filtering problems under multiple exogenous disturbances. There are N potentially possible disturbance inputs of a system on each of which may act a disturbance from a certain class. A disturbance attenuation level is defined for each channel as an induced norm of the operator mapping signals of the corresponding class to the objective output of the system. Necessary conditions of the Pareto optimality in the multi-objective control or filtering problems are derived. A multi-objective cost parameterized by weights from an N-dimensional simplex is introduced for the disturbances acting on all inputs simultaneously. It is established that the optimal solutions with respect to this cost are the Pareto suboptimal solutions and their relative losses compared to the Pareto optimal ones do not exceed 1 − √N/N. These results are extended to the case when the disturbances acting on different inputs are combined into coalitions.

Acid pre-treatment of bagasse is performed in a batch reactor, in which concentration of components evolve over time. This type of systems is usually studied using optimal control, which is concerned with dynamic systems and their optimization over time. This study applies the theory of optimal control for multi-objective optimization. The objectives include maximization of xylose and acid soluble lignin (ASL) fraction, which is obtained during acid pre-treatment. The Pontryagin's maximum principle in combination with the numerical method of steepest descent of Hamiltonian is used to solve multi-objective optimal control problem. The results showed that maximum xylose and acid soluble lignin (ASL) yield was obtained around 0.79 (g/g of xylan) and 0.3 (g/g of lignin), respectively, at final time (t <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> =300 min.). However, acid pre-treatment is influenced by various sources of uncertainties. Therefore, it is essential to optimize the acid pre-treatment process after considering these variabilities. In this work, variability in feedstock composition has been considered, and the stochastic maximum principle is used to perform stochastic multi-objective optimal control problem. Deterministic and stochastic control profiles were obtained for different weights. An open loop simulation of Ito model, using deterministic and stochastic control profile, was performed for 1000 samples and statistics were generated by assuming normal distribution with 95% confidence bound.