Bound-constrained problems constraints

Problem 4.1 is nonlinear if one or more of the functions/, gv...,gm are nonlinear. It is unconstrained if there are no constraint functions g, and no bounds on the jc,., and it is bound-constrained if only the xt are bounded. In linearly constrained problems all constraint functions g, are linear, and the objective/is nonlinear. There are special NLP algorithms and software for unconstrained and bound-constrained problems, and we describe these in Chapters 6 and 8. Methods and software for solving constrained NLPs use many ideas from the unconstrained case. Most modem software can handle nonlinear constraints, and is especially efficient on linearly constrained problems. A linearly constrained problem with a quadratic objective is called a quadratic program (QP). Special methods exist for solving QPs, and these iare often faster than general purpose optimization procedures. 

Some well-known stochastic methods for solving SOO problems are simulated annealing (SA), GAs,DE and particle swarm optimization (PSO). These were initially proposed and developed for optimization problems with bounds only [that is, unconstrained problems without Equations (4.7) and (4.8)]. Subsequently, they were extended to constrained problems by incorporating a strategy for handling constraints. One relatively simple and popular sdategy is the penalty function, which involves modifying the objective function (Equation 4.5) by the addition (in the case of minimization) of a term which depends on constraint violation. Eor example, see Equation (4.9),... 

This approach operates in two phases. First, a sufficient number of elements is found in order to satisfy the linearization of all of the constraints at the initial point. In this way we guarantee that a feasible QP subproblem exists for (27). Second, to avoid convergence to a suboptimal solution with too few elements, we retain additional dummy elements in the formulation that are constrained to be less than or equal to a negligible element length. These elements can be placed at all nonzero element locations, but in practice they need only be associated with elements that have active error bounds at the QP solution. Now once the QP subproblem is solved, multipliers on the upper bounds of the dummy elements are checked for positive values. These indicate that the objective function can be further improved by relaxing the dummy element. After relaxation (which effectively adds another nonzero element to the problem), another dummy element is added in order to allow for any additional nonzero elements that may be needed. 

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 online solution of this constrained estimation problem, known as full information estimator because we consider all the available measurements, is formulated as an optimization problem - typically posed as a least squares mathematical program-subject to the model constraints and inequality constraints that represents bounds on variables or equations. 

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. 

Constrained optimization is broached starting from Chapter 9. The constraints are split into three categories bounds, equality constraints, and inequality constraints. The relationship between primal and dual problems is discussed in further depth. 

Nominal design values of the spatial structure of the column for the synthesis of MTBE (i.e. column diameter and reboiler and condenser heat exchange areas) are firstly estimated. They are obtained by solving a steady-state optimization problem, which minimizes the total annualized cost of the RD unit in the absence of disturbances. The following constraints are included to the problem formulation (i) the column diameter is bounded by flooding conditions (ii) the heat exchange areas of the condenser and reboiler are estimated by bounded values of outlet temperatures of the hot and cold utilities, and (m) the molar fraction of MTBE at the top and bottom stream is constrained to values lower than 0.1% and 99% respectively. Thus, the statics optimization problem results in,... 

The third problem is also common especially for mechanism with constraints. If control points to generate desired motion are found, it is not guaranteed that they can be realized. Reachable area of constrained mechanism is bounded. That of beam-shaped gel uniformly coupled by electric fields was thoroughly studied... 

The formulation of the s-constraint technique is performed as one of the objectives is assigned as the objective function while the others are constrained within specified upper limits. The selected process parameters are assigned as the decision variables of the optimisation problem. The optimiser searches over the process variables, within the feasibility and constraints regions and feeds these selected variables to the model in HYSYS. Then, it waits for the process in HYSYS to converge and then recalculate the objectives and evaluate the optimisation results. This search loop between the optimiser in Excel and the model in HYSYS continues until a global optimum point is found which represents a point on the Pareto curve. The above optimisation process is repeated for different bounds of the constrained objectives to develop the entire Pareto curve. 

The algorithmic steps for the constrained aBB approach can be generalized to any force field model or routine for solving constrained optimization problems. Here, the otBB approach is interfaced with PACK [74] and NPSOL [28]. PACK is used to transform to and from Cartesian and internal coordinate systems, as well as to obtain function and gradient contributions for the ECEPP/3 force field and the distance constraint equations. NPSOL is a local nonlinear optimization solver that is used to locally solve the constrained upper and lower bounding problems in each subdomain. 

The modified constrained global optimization was also applied to the Compstatin structure prediction problem using the same constraint function and parameters [104]. The goal of introducing TAD as a component of the upper bound solution approach is to increase the number of feasible points available for initialization of the constrained local minimization. Initially, TAD is used in combination with simple van der Waals overlap restraints to drive the distance violations to zero. 

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. 

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