Constrained nonlinear optimization

Klein, M. and R. R. Kimpel. Application of Linearly Constrained Nonlinear Optimization to Plant to Location and Sizing J Indus Engin 18 90 (1967). 

We solve the nonlinear formulation of the semidefinite program by the augmented Lagrange multiplier method for constrained nonlinear optimization [28, 29]. Consider the augmented Lagrangian function... 

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. 

The Lagrange multipliers in a constrained nonlinear optimization problem have a similar interpretation to the dual variables or shadow prices in linear programming. To provide such an interpretation, we will consider problem (3.3) with only equality constraints that is,... 

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. 

This chapter introduces the fundamentals of unconstrained and constrained nonlinear optimization. Section 3.1 presents the formulation and basic definitions of unconstrained nonlinear optimization along with the necessary, sufficient, and necessary and sufficient optimality conditions. For further reading refer to Hestenes (1975), Luenberger (1984), and Minoux (1986). 

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. 

There are essentially six types of procedures to solve constrained nonlinear optimization problems. The three methods considered more successful are the successive LP, the successive quadratic programming, and the generalized reduced gradient method. These methods use different strategies but the same information to move from a starting point to the optimum, the first partial derivatives of the economic model, and constraints evaluated at the current point. Successive LP is used in a number of solvers including MINOS. Successive quadratic programming is the method of... 

Problem Type Unconstrained and constrained nonlinear optimization Method Various methods... 

Agrawal, A., Saran, A.D., Rath, S.S., Khanna, A., 2004. Constrained nonlinear optimization for solubility parameters of poly(lactic acid) and poly(glycolic acid)- validation and comparison. Polymer 45, 8603-8612. 

Using the constrained nonlinear optimization method, the optimal test plan is... 

In view of the dramatic decreases in the ratio of computer cost to performance in recent years, it can be argued that physically based, nonlinear process models should be used in the set-point calculations, instead of approximate linear models. However, linear models are still widely used in MPC applications for three reasons First, linear models are reasonably accurate for small changes in u and d and can easily be updated based on current data or a physically based model. Second, some model inaccuracy can be tolerated, because the calculations are repeated on a frequent basis and they include output feedback from the measurements. Third, the computational effort required for constrained, nonlinear optimization is still relatively large, but is decreasing. 

The goal of an optimization problem is to find a vector p in the search space S so that certain quality criterion is satisfied, namely, the error norm g(p) in Eq. 14 is minimized. The vector p is a solution to the minimization problem if g(p ) is a global minimum in S. For the constrained nonlinear optimization problem associated with the identification of differential hysteresis, the error surface defined by the objective function and constraints can exhibit many local minima or can even be multimodal. For this reason, different solution techniques will have dramatically different performance. The primary consideration in evaluating an optimization algorithm may be convergence speed or the minimum error achieved. Secondary consideration may be consistency, robustness, computational efficiency, or tracking capabilities. 

The area of nonlinear optimization can be subdivided further into various classes of problems. Firstly, there is constrained versus unconstrained optimization. Normally, the inclusion of constraints generates only a special case for the sibling, unconstrained... 

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. 

Zhou, J., Tits, A., Lawrence, C. Users s Guide for FFSQP Version 3.7 A FORTRAN code for solving constrained nonlinear (minimax) optimization problems, generating iterates satisfying all inequality and linear constraints, University of Maryland, 1997. 

Wendt M., Li P. and Wozny G. 2002. Nonlinear chance constrained process optimization under uncertainty, Ind. Eng. Chem. Res., 41, 3621-3629. 

Optimization problems in which objective function and/or constrains are not linear define nonlinear optimization problems. While linear optimization problems can be solved in polynomial time, generally nonlinear optimization problems are much more difficult to solve. In discrete optimization problems, variables are defined discrete and thus they are nmilinear optimization problems. 

Find (a feasible) X which maximizes/mini-mizes the objective function f(X) subject to the given constraints g(X) < > = b. Optimization problems, in which at least one of the functions among objectives and constrains is nonlinear, define a nonlinear optimization problem. This type of problem is the most general one, and all other problems can be considered as special cases of the nonlinear programming problem (Rao 2009). 

Nonlinear optimization problems in just a single decision variable frequently arise in chemical engineering applications. If the objective function is unconstrained, the optimal solutionis) can often be obtained analytically using derivatives from calculus. When constrained, numerical methods are frequently necessary. Some applications that have appeared often in chemical engineering textbooks include the following, many of which involve a balance between capital and operating costs ... 

Chapter 13 illustrates the problem of constrained optimization by introducing the active set methods. Successive linear programming (SLP), projection, reduced direction search, SQP methods are described, implemented, and adopted to solve several practical examples of constrained linear/nonlinear optimization, including the solution of the Maratos effect. 

As mentioned earlier, the developed algorithm employs dynopt to solve the intermediate problems associated with the local interaction of the agents. Specifically, dynopt is a set of MATLAB functions that use the orthogonal collocation on finite elements method for the determination of optimal control trajectories. As inputs, this toolbox requires the dynamic process model, the objective function to be minimized, and the set of equality and inequality constraints. The dynamic model here is described by the set of ordinary differential equations and differential algebraic equations that represent the fermentation process model. For the purpose of optimization, the MATLAB Optimization Toolbox, particularly the constrained nonlinear rninimization routine fmincon [29], is employed. 

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