scholarly journals On a Class of Second-Order PDE&PDI Constrained Robust Modified Optimization Problems

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1473
Author(s):  
Savin Treanţă
Keyword(s):  
Optimization Problems ◽  
Integral Functional ◽  
Optimal Solution ◽  
Multiple Integral ◽  
Second Order ◽  
Control Problems ◽  
Mathematical Framework ◽  
Mixed Constraints ◽  
Scalar Multiple ◽  
Cost Functionals

In this paper, by using scalar multiple integral cost functionals and the notion of convexity associated with a multiple integral functional driven by an uncertain multi-time controlled second-order Lagrangian, we develop a new mathematical framework on multi-dimensional scalar variational control problems with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Concretely, we introduce and investigate an auxiliary (modified) variational control problem, which is much easier to study, and provide some equivalence results by using the notion of a normal weak robust optimal solution.

scholarly journals On Robust Saddle-Point Criterion in Optimization Problems with Curvilinear Integral Functionals

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1790
Author(s):  
Savin Treanţă ◽  
Koushik Das
Keyword(s):  
Saddle Point ◽  
Optimization Problems ◽  
Integral Functional ◽  
Optimal Solution ◽  
Control Problems ◽  
New Class ◽  
Mixed Constraints ◽  
Curvilinear Integral ◽  
Lagrange Functional ◽  
Cost Functionals

In this paper, we introduce a new class of multi-dimensional robust optimization problems (named (P)) with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Moreover, we define an auxiliary (modified) class of robust control problems (named (P)(b¯,c¯)), which is much easier to study, and provide some characterization results of (P) and (P)(b¯,c¯) by using the notions of normal weak robust optimal solution and robust saddle-point associated with a Lagrange functional corresponding to (P)(b¯,c¯). For this aim, we consider path-independent curvilinear integral cost functionals and the notion of convexity associated with a curvilinear integral functional generated by a controlled closed (complete integrable) Lagrange 1-form.

scholarly journals Second-Order PDE Constrained Controlled Optimization Problems with Application in Mechanics

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1472
Author(s):  
Savin Treanţă
Keyword(s):  
Control Problem ◽  
Necessary Conditions ◽  
Optimization Problems ◽  
Multiple Integral ◽  
Second Order ◽  
Final Part ◽  
Control Problems ◽  
Partial Derivatives ◽  
The One ◽  
Cost Functionals

The present paper deals with a class of second-order PDE constrained controlled optimization problems with application in Lagrange–Hamilton dynamics. Concretely, we formulate and prove necessary conditions of optimality for the considered class of control problems driven by multiple integral cost functionals involving second-order partial derivatives. Moreover, an illustrative example is provided to highlight the effectiveness of the results derived in the paper. In the final part of the paper, we present an algorithm to summarize the steps for solving a control problem such as the one investigated here.

scholarly journals Duality Theorems for (ρ,ψ,d)-Quasiinvex Multiobjective Optimization Problems with Interval-Valued Components

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 894
Author(s):  
Savin Treanţă
Keyword(s):  
Multiobjective Optimization ◽  
Optimization Problems ◽  
Integral Functional ◽  
Multiple Integral ◽  
Control Problems ◽  
Duality Theorems ◽  
New Class ◽  
Duality Results ◽  
Multiobjective Optimization Problems ◽  
Interval Valued

The present paper deals with a duality study associated with a new class of multiobjective optimization problems that include the interval-valued components of the ratio vector. More precisely, by using the new notion of (ρ,ψ,d)-quasiinvexity associated with an interval-valued multiple-integral functional, we formulate and prove weak, strong, and converse duality results for the considered class of variational control problems.

scholarly journals A certified RB method for PDE-constrained parametric optimization problems

2019 ◽  
Vol 10 (1) ◽  
pp. 123-152
Author(s):  
Andrea Manzoni ◽  
Stefano Pagani
Keyword(s):  
Optimal Control ◽  
Efficient Solution ◽  
Parametric Optimization ◽  
Optimization Problems ◽  
A Posteriori Error Estimates ◽  
Optimal Solution ◽  
Posteriori Error ◽  
Elliptic Pdes ◽  
Reduced Basis ◽  
Cost Functionals

Abstract We present a certified reduced basis (RB) framework for the efficient solution of PDE-constrained parametric optimization problems. We consider optimization problems (such as optimal control and optimal design) governed by elliptic PDEs and involving possibly non-convex cost functionals, assuming that the control functions are described in terms of a parameter vector. At each optimization step, the high-fidelity approximation of state and adjoint problems is replaced by a certified RB approximation, thus yielding a very efficient solution through an “optimize-then-reduce” approach. We develop a posteriori error estimates for the solutions of state and adjoint problems, the cost functional, its gradient and the optimal solution. We confirm our theoretical results in the case of optimal control/design problems dealing with potential and thermal flows.

scholarly journals Smoothing approximations to nonsmooth optimization problems

1995 ◽  
Vol 36 (3) ◽  
pp. 274-285 ◽  
Author(s):  
X.Q. Yang
Keyword(s):  
Optimization Problems ◽  
Directional Derivative ◽  
Optimal Solution ◽  
Necessary Optimality Conditions ◽  
Second Order ◽  
Smooth Approximation ◽  
Minimization Problems ◽  
Second Order Directional Derivative ◽  
Nonsmooth Minimization ◽  
Original Objective

AbstractWe study certain types of composite nonsmooth minimization problems by introducing a general smooth approximation method. Under various conditions we derive bounds on error estimates of the functional values of original objective function at an approximate optimal solution and at the optimal solution. Finally, we obtain second-order necessary optimality conditions for the smooth approximation prob lems using a recently introduced generalized second-order directional derivative.

scholarly journals A Survey on Low-Thrust Trajectory Optimization Approaches

Aerospace ◽  
2021 ◽  
Vol 8 (3) ◽  
pp. 88
Author(s):  
David Morante ◽  
Manuel Sanjurjo Rivo ◽  
Manuel Soler
Keyword(s):  
Optimal Control ◽  
Objective Function ◽  
Trajectory Optimization ◽  
Optimal Control Problems ◽  
Optimization Problems ◽  
Control Problems ◽  
Mathematical Framework ◽  
Low Thrust ◽  
Discrete State ◽  
Hybrid Optimal Control

In this paper, we provide a survey on available numerical approaches for solving low-thrust trajectory optimization problems. First, a general mathematical framework based on hybrid optimal control will be presented. This formulation and their elements, namely objective function, continuous and discrete state and controls, and discrete and continuous dynamics, will serve as a basis for discussion throughout the whole manuscript. Thereafter, solution approaches for classical continuous optimal control problems will be briefly introduced and their application to low-thrust trajectory optimization will be discussed. A special emphasis will be placed on the extension of the classical techniques to solve hybrid optimal control problems. Finally, an extensive review of traditional and state-of-the art methodologies and tools will be presented. They will be categorized regarding their solution approach, the objective function, the state variables, the dynamical model, and their application to planetocentric or interplanetary transfers.

scholarly journals On Well-Posedness of Some Constrained Variational Problems

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2478
Author(s):  
Savin Treanţă
Keyword(s):  
Variational Inequality ◽  
Lower Semicontinuity ◽  
Integral Functional ◽  
Inequality Constraints ◽  
Multiple Integral ◽  
Variational Problems ◽  
Well Posedness ◽  
New Class ◽  
Scalar Multiple ◽  
Constrained Variational Problems

By considering the new forms of the notions of lower semicontinuity, pseudomonotonicity, hemicontinuity and monotonicity of the considered scalar multiple integral functional, in this paper we study the well-posedness of a new class of variational problems with variational inequality constraints. More specifically, by defining the set of approximating solutions for the class of variational problems under study, we establish several results on well-posedness.

scholarly journals A Noether Theorem on Unimprovable Conservation Laws for Vector-Valued Optimization Problems in Control Theory

2006 ◽  
Vol 13 (1) ◽  
pp. 173-182 ◽  
Author(s):  
Delfim F. M. Torres
Keyword(s):  
Optimal Control ◽  
Optimal Control Problems ◽  
Optimization Problems ◽  
Control Problems ◽  
Control Engineering ◽  
Mathematical Question ◽  
Isoperimetric Constraints ◽  
Cost Functionals ◽  
Vector Valued ◽  
Invariance Theorem

Abstract We obtain a version of Noether's invariance theorem for optimal control problems with a finite number of cost functionals. The result is obtained by formulating E. Noether's result for optimal control problems subject to isoperimetric constraints, and then using the unimprovable (Pareto) notion of optimality. It was A. Gugushvili who drew the author's attention to a result of this kind that was posed as an open mathematical question of a great interest in applications of control engineering.

scholarly journals Well Posedness of New Optimization Problems with Variational Inequality Constraints

2021 ◽  
Vol 5 (3) ◽  
pp. 123
Author(s):  
Savin Treanţă
Keyword(s):  
Variational Inequality ◽  
Lower Semicontinuity ◽  
Optimization Problems ◽  
Integral Functional ◽  
Inequality Constraints ◽  
Multiple Integral ◽  
Well Posedness ◽  
Constrained Optimization Problems ◽  
New Class ◽  
Theoretical Developments

In this paper, we studied the well posedness for a new class of optimization problems with variational inequality constraints involving second-order partial derivatives. More precisely, by using the notions of lower semicontinuity, pseudomonotonicity, hemicontinuity and monotonicity for a multiple integral functional, and by introducing the set of approximating solutions for the considered class of constrained optimization problems, we established some characterization results on well posedness. Furthermore, to illustrate the theoretical developments included in this paper, we present some examples.

scholarly journals APPLICATION OF THE ”WAVE“ METHOD TO SOLVE THE PROBLEMS OF OPTIMIZATION OF PASSENGER AND CARGO TRANSPORTATION IN MEGACITIES TO IMPROVE THEIR TRANSPORT NETWORK AND INFRASTRUCTURE. Conceptual basis, mathematical framework

2021 ◽  
pp. 410-427
Author(s):  
Yuriy Chovnyuk ◽  
Katerina Razumova ◽  
Petro Cherednichenko ◽  
Olena Mischenko
Keyword(s):  
Optimization Problems ◽  
Integral Functional ◽  
Classical Problem ◽  
Variational Calculus ◽  
Optimal Location ◽  
Physical Optics ◽  
Mathematical Framework ◽  
Wave Method ◽  
Global Extremum ◽  
Transport Logistics

The paper proposes a new approach to solving optimization problems arising in engineering and transport logistics in designing and construction of roads (in particular, in megacities, near large transport hubs, near state borders) for cargo and passenger transportation and implementation of international trade. The fundamental problems of modern engineering logistics - the problem of optimal location (transport hubs) and the problem of identification and segmentation of logistics, transport and logistics zones are considered. These problems are solved using methods of variational calculus, in particular, the so-called "wave" method based on the Fermat principle existing in physical optics, which is based on the analogy between finding the global extremum of the integral functional and the propagation of light in an optically heterogeneous medium.  A numerical method for the above technique has been developed programatically. The idea of the "wave method/approach belongs to V.V. Bashurov, who proposed to use the methods of geometrical and physical optics to investigate applied safety problems and some related issues. The essence of the "wave method" is that initially the safety problem is reduced to the search for the global minimum of a nonlinear functional. In turn, the minimization problem is solved by constructing the trajectory of motion of the front of the "light wave" moving in an optically inhomogeneous medium. Finding the minimum of a functional is a classical problem of variational calculus, for the solution of which a significant mathematical apparatus has been developed. However, most of known methods effectively determine only local extrema. "Wave" method allows to solve the problem of finding a global extremum with greater efficiency. This paper proposes a conceptual framework and scientifically justified modification of this "wave" method for solving optimization problems arising in engineering and transport logistics, including the problem of optimal location of the transport hub, transport and logistics center (warehouse) and the problem of optimal identification and segmentation of logistical zones (metropolitan areas, large transport hubs).

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