Deduction of the Classical Optimality Conditions in Nonlinear Optimization

1997 ◽  
pp. 45-60
Author(s):  
Tamás Rapcsák
Keyword(s):  
Optimality Conditions ◽  
Nonlinear Optimization

Fundamentals of Nonlinear Optimization

1995 ◽  
Author(s):  
Christodoulos A. Floudas
Keyword(s):  
Optimality Conditions ◽  
Nonlinear Optimization ◽  
Optimization Problems ◽  
Nonlinear Function ◽  
Nonlinear Least Squares ◽  
Sufficient Optimality Conditions ◽  
Science And Engineering ◽  
Chemical Phase ◽  
Unconstrained Nonlinear Optimization ◽  
Necessary And Sufficient

This chapter discusses the fundamentals of nonlinear optimization. Section 3.1 focuses on optimality conditions for unconstrained nonlinear optimization. Section 3.2 presents the first-order and second-order optimality conditions for constrained nonlinear optimization problems. This section presents the formulation and basic definitions of unconstrained nonlinear optimization along with the necessary, sufficient, and necessary and sufficient optimality conditions. An unconstrained nonlinear optimization problem deals with the search for a minimum of a nonlinear function f(x) of n real variables x = (x1, x2 , . . . , xn and is denoted as Each of the n nonlinear variables x1, x2 , . . . , xn are allowed to take any value from - ∞ to + ∞. Unconstrained nonlinear optimization problems arise in several science and engineering applications ranging from simultaneous solution of nonlinear equations (e.g., chemical phase equilibrium) to parameter estimation and identification problems (e.g., nonlinear least squares).

On second-order optimality conditions in nonlinear optimization

2016 ◽  
Vol 32 (1) ◽  
pp. 22-38 ◽  
Author(s):  
Roberto Andreani ◽  
Roger Behling ◽  
Gabriel Haeser ◽  
Paulo J.S. Silva
Keyword(s):  
Optimality Conditions ◽  
Nonlinear Optimization ◽  
Second Order ◽  
Second Order Optimality Conditions

scholarly journals A Theoretical Framework for Optimality Conditions of Nonlinear Type-2 Interval-Valued Unconstrained and Constrained Optimization Problems Using Type-2 Interval Order Relations

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 908
Author(s):  
Md Sadikur Rahman ◽  
Ali Akbar Shaikh ◽  
Irfan Ali ◽  
Asoke Kumar Bhunia ◽  
Armin Fügenschuh
Keyword(s):  
Constrained Optimization ◽  
Optimality Conditions ◽  
Nonlinear Optimization ◽  
Optimization Problems ◽  
Interval Order ◽  
Constrained Optimization Problems ◽  
Interval Approach ◽  
Kkt Optimality Conditions ◽  
Interval Valued

In the traditional nonlinear optimization theory, the Karush-Kuhn-Tucker (KKT) optimality conditions for constrained optimization problems with inequality constraints play an essential role. The situation becomes challenging when the theory of traditional optimization is discussed under uncertainty. Several researchers have discussed the interval approach to tackle nonlinear optimization uncertainty and derived the optimality conditions. However, there are several realistic situations in which the interval approach is not suitable. This study aims to introduce the Type-2 interval approach to overcome the limitation of the classical interval approach. This study introduces Type-2 interval order relation and Type-2 interval-valued function concepts to derive generalized KKT optimality conditions for constrained optimization problems under uncertain environments. Then, the optimality conditions are discussed for the unconstrained Type-2 interval-valued optimization problem and after that, using these conditions, generalized KKT conditions are derived. Finally, the proposed approach is demonstrated by numerical examples.

Optimal control of higher order differential inclusions with functional constraints

2020 ◽  
Vol 26 ◽  
pp. 37 ◽  
Author(s):  
Elimhan N. Mahmudov
Keyword(s):  
Optimal Control ◽  
Optimality Conditions ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Higher Order ◽  
Second Order ◽  
Sufficient Optimality Conditions ◽  
Functional Constraints ◽  
Complementary Slackness ◽  
Complementary Slackness Conditions

The present paper studies the Mayer problem with higher order evolution differential inclusions and functional constraints of optimal control theory (PFC); to this end first we use an interesting auxiliary problem with second order discrete-time and discrete approximate inclusions (PFD). Are proved necessary and sufficient conditions incorporating the Euler–Lagrange inclusion, the Hamiltonian inclusion, the transversality and complementary slackness conditions. The basic concept of obtaining optimal conditions is locally adjoint mappings and equivalence results. Then combining these results and passing to the limit in the discrete approximations we establish new sufficient optimality conditions for second order continuous-time evolution inclusions. This approach and results make a bridge between optimal control problem with higher order differential inclusion (PFC) and constrained mathematical programming problems in finite-dimensional spaces. Formulation of the transversality and complementary slackness conditions for second order differential inclusions play a substantial role in the next investigations without which it is hardly ever possible to get any optimality conditions; consequently, these results are generalized to the problem with an arbitrary higher order differential inclusion. Furthermore, application of these results is demonstrated by solving some semilinear problem with second and third order differential inclusions.

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