Optimality conditions constrained problems

V L is equal to the constrained derivatives for the problem, which should be zero at the solution to the problem. Also, these stationarity conditions very neatly provide the necessaiy conditions for optimality of an equality-constrained problem. 

Remark 2 Gordon s theorem has been frequently used in the derivation of optimality conditions of nonlinearly constrained problems. 

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 first the formulation and basic definitions of constrained nonlinear optimization problems and introduces the Lagrange function and the Lagrange multipliers along with their interpretation. Subsequently, the Fritz John first-order necessary optimality conditions are discussed as well as the need for first-order constraint qualifications. Finally, the necessary, sufficient Karush-Kuhn-Dicker conditions are introduced along with the saddle point necessary and sufficient optimality conditions. 

A key idea in developing necessary and sufficient optimality conditions for nonlinear constrained optimization problems is to transform them into unconstrained problems and apply the optimality conditions discussed in Section 3.1 for the determination of the stationary points of the unconstrained function. One such transformation involves the introduction of an auxiliary function, called the Lagrange function L(x,A, p), defined as... 

In this section we present, under the assumption of differentiability, the first-order necessary optimality conditions for the constrained nonlinear programming problem (3.3) as well as the corresponding geometric necessary optimality conditions. 

The optimality conditions discussed in the previous sections formed the theoretical basis for the development of several algorithms for unconstrained and constrained nonlinear optimization problems. In this section, we will provide a brief outline of the different classes of nonlinear multivariable optimization algorithms. 

Part 1, comprised of three chapters, focuses on the fundamentals of convex analysis and nonlinear optimization. Chapter 2 discusses the key elements of convex analysis (i.e., convex sets, convex and concave functions, and generalizations of convex and concave functions), which are very important in the study of nonlinear optimization problems. Chapter 3 presents the first and second order optimality conditions for unconstrained and constrained nonlinear optimization. Chapter 4 introduces the basics of duality theory (i.e., the primal problem, the perturbation function, and the dual problem) and presents the weak and strong duality theorem along with the duality gap. Part 1 outlines the basic notions of nonlinear optimization and prepares the reader for Part 2. 

In an attempt to avoid the ill-conditioning that occurs in the regular pentilty tuid bturier function methods, Hestenes (1969) and PoweU (1969) independently developed a multiplier method for solving nonhnearly constrained problems. This multiplier method was originally developed for equality constraints and involves optimizing a sequence of unconstrained augmented Lagrtuigitui functions. It was later extended to handle inequality constraints by Rockafellar (1973). 

Each of the inequality constraints gj(z) multiphed by what is called a Kuhn-Tucker multiplier is added to form the Lagrange function. The necessaiy conditions for optimality, called the Karush-Kuhn-Tucker conditions for inequality-constrained optimization problems, are... 

The operation of a plant under steady-state conditions is commonly represented by a non-linear system of algebraic equations. It is made up of energy and mass balances and may include thermodynamic relationships and some physical behavior of the system. In this case, data reconciliation is based on the solution of a nonlinear constrained optimization problem. 

In many crystallographic problems, the choice of the variables x is subject to constraints (boundary conditions represented by equations). The problem is then known as a constrained optimization problem. An example would be the refinement of a... 

A typical SQP termination condition for a constrained optimization problem... 

In summary, condition 1 gives a set of n algebraic equations, and conditions 2 and 3 give a set of m constraint equations. The inequality constraints are converted to equalities using h slack variables. A total of M + m constraint equations are solved for n variables and m Lagrange multipliers that must satisfy the constraint qualification. Condition 4 determines the value of the h slack variables. This theorem gives an indirect problem in which a set of algebraic equations is solved for the optimum of a constrained optimization problem. 

---