Equality unconstrained problem

Lagrange discovered how to transform a constrained optimization problem, with equality constraints, into an unconstrained problem. To solve the problem... 

Alternatively, increasing penalties may be applied on constraint violations during repeated applications of any computational algorithm used for unconstrained problems. We will use the latter approach in Chapter 7 to solve optimal control problems constrained by (in)equalities. 

Lastly, it is also possible to use the method that exploits the null space of the constraints. Once again in this case, all the active bounds must first be removed from the problem. Only then it is possible to use either LQ factorization or a stable Gauss factorization of all the equality and active inequality constraints. This gives the KKT conditions for an unconstrained problem, as has already been demonstrated for equality constraints. 

In problems in which there are n variables and m equality constraints, we could attempt to eliminate m variables by direct substitution. If all equality constraints can be removed, and there are no inequality constraints, the objective function can then be differentiated with respect to each of the remaining (n — m) variables and the derivatives set equal to zero. Alternatively, a computer code for unconstrained optimization can be employed to obtain x. If the objective function is convex (as in the preceding example) and the constraints form a convex region, then any stationary point is a global minimum. Unfortunately, very few problems in practice assume this simple form or even permit the elimination of all equality constraints. 

Despite the exactness feature of Pv no general-purpose, widely available NLP solver is based solely on the Lx exact penalty function Pv This is because Px also has a negative characteristic it is nonsmooth. The term hj(x) has a discontinuous derivative at any point x where hj (x) = 0, that is, at any point satisfying the y th equality constraint in addition, max 0, gj (x) has a discontinuous derivative at any x where gj (x) = 0, that is, whenever the yth inequality constraint is active, as illustrated in Figure 8.6. These discontinuities occur at any feasible or partially feasible point, so none of the efficient unconstrained minimizers for smooth problems considered in Chapter 6 can be applied, because they eventually encounter points where Px is nonsmooth. 

GRG Probably most robust of all three methods Versatile—especially good for unconstrained or linearly constrained problems but also works well for nonlinear constraints Once it reaches a feasible solution it remains feasible and then can be stopped at any stage with an improved solution Needs to satisfy equalities at each step of the algorithm... 

Unconstrained Optimization Unconstrained optimization refers to the case where no inequality constraints are present and all equality constraints can be eliminated by solving for selected dependent variables followed by substitution for them in the objective function. Very few realistic problems in process optimization are unconstrained. However, the availability of efficient unconstrained optimization techniques is important because these techniques must be applied in real time, and iterative calculations may require excessive computer time. Two classes of unconstrained techniques are single-variable optimization and multivariable optimization. 

In this approach, the process variables are partitioned into dependent variables and independent variables (optimisation variables). For each choice of the optimisation variables (sometimes referred to as decision variables in the literature) the simulator (model solver) is used to converge the process model equations (described by a set of ODEs or DAEs). Therefore, the method includes two levels. The first level performs the simulation to converge all the equality constraints and to satisfy the inequality constraints and the second level performs the optimisation. The resulting optimisation problem is thus an unconstrained nonlinear optimisation problem or a constrained optimisation problem with simple bounds for the associated optimisation variables plus any interior or terminal point constraints (e.g. the amount and purity of the product at the end of a cut). Figure 5.2 describes the solution strategy using the feasible path approach. 

The conditions yielding the unconstrained maximum centerline deposition rate give a deposition uniformity of only about 25%. While this may well be acceptable for some fiber coating processes, there are likely applications for which it is not. We now consider the problem of maximizing the centerline deposition rate, subject to an additional constraint that the deposition uniformity satisfies some minimum requirement. Assuming that the required uniformity is better than that obtained in the unconstrained case, the constrained maximum centerline deposition rate should occur when the uniformity constraint is just marginally satisfied. This permits replacing the inequality constraint of a minimum uniformity by an equality constraint that is satisfied exactly. 

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). 

The simplest optimization problems are those without equality constraints, inequality constraints, and lower and upper bounds. They are referred to as unconstrained optimization. Otherwise, if one or more constraints apply, the problem is one in constrained optimization. 

This is a typical minimization problem for a function of n variables that can be solved using a Mathcad built-in function MINIMIZE. The latter implements gradient search algorithms to find the local minimum. The SSq function in this case is called the target function, and the unknown kinetic constants are the optimization parameters. When there are no additional limitations for the values of optimization parameters or the sought function, we have a case of the so called unconstrained optimization. Likewise, if the unknown parameters or the target function itself are mathematically constrained with some equalities or inequalities, then one deals with the constrained optimization. Such additional constrains are usually set on the basis on the physical nature of the problems (e.g. rate constants must be positive, a ratio of the direct reaction rate to that of the inverse one must equal the equilibrium constant, etc.) The constraints are sometimes added in order to speed up the computations (for example, the value of target function in the found minimum should not exceed some number TOL). 

Optimization This refers to minimizing or maximizing a real-valued function f(x). The permitted values for x = (xj,..., xj can be either constrained or unconstrained. The linear programming problem is a well-known and important case f(x) is linear, and there are linear equality and/or inequality constraints on x. 

Unconstrained optimization refers to the situation where there are no inequality constraints and all equality constraints can be eliminated by variable substitution in the objective function. First we consider single-variable optimization, followed by optimization problems with multiple variables. Because optimization techniques are iterative in nature, we focus mainly on efficient methods that can be applied on-line. Most RTO applications are multivariable... 

The methods for solving an optimization task depend on the problem classification. Since the maximum of a function / is the minimum of the function —/, it suffices to deal with minimization. The optimization problem is classified according to the type of independent variables involved (real, integer, mixed), the number of variables (one, few, many), the functional characteristics (linear, least squares, nonlinear, nondifferentiable, separable, etc.), and the problem. statement (unconstrained, subject to equality constraints, subject to simple bounds, linearly constrained, nonlinearly constrained, etc.). For each category, suitable algorithms exist that exploit the problem s structure and formulation. 

An optimization problem may be unconstrained, in which case each xj can take any real value, or it can be constrained, such that an allowable jc must satisfy some collection of equality and inequality constraints... 

In the preceding sections, we considered only unconstrained optimization problems in which X may take any value. Here, we extend these methods to constrained minimization problems, where to be acceptable (or feasible), x must satisfy a number e of equality constraints gi (x) = 0 and a number n of inequality constraints hj x) > 0, where each g, (x) and hj(x) are assumed to be differentiable nonlinear functions. This constrained optimization problem... 

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