Constrained feasible region

Suppose you are given the task of preparing a ternary (three-component) solvent system such that the total volume be 1.00 liter. Write the equality constraint in terms of x X2, and Xj, the volumes of each of the three solvents. Sketch the three-dimensional factor space and clearly draw within it the planar, two-dimensional constrained feasible region. (Hint try a cube and a triangle after examining Figure 2.16.)... 

It is also worth noting that the stochastic optimization methods described previously are readily adapted to the inclusion of constraints. For example, in simulated annealing, if a move suggested at random takes the solution outside of the feasible region, then the algorithm can be constrained to prevent this by simply setting the probability of that move to 0. 

The perceptional advantages of response contours in illustrating nonlinear blending behavior and the additional information of the experimental boundary locations were incorporated into a generalized algorithm which determines the feasible region on a tricoordinate plot for a normal or pseudocomponent mixture having any number of constrained components. 

Neither of the problems illustrated in Figures 4.5 and 4.6 had more than one optimum. It is easy, however, to construct nonlinear programs in which local optima occur. For example, if the objective function / had two minima and at least one was interior to the feasible region, then the constrained problem would have two local minima. Contours of such a function are shown in Figure 4.7. Note that the minimum at the boundary point x1 = 3, x2 = 2 is the global minimum at / = 3 the feasible local minimum in the interior of the constraints is at / = 4. 

Geometry of a constrained optimization problem. The feasible region lies within the binding constraints plus the boundaries themselves. 

Like penalty methods, barrier methods convert a constrained optimization problem into a series of unconstrained ones. The optimal solutions to these unconstrained subproblems are in the interior of the feasible region, and they converge to the constrained solution as a positive barrier parameter approaches zero. This approach contrasts with the behavior of penalty methods, whose unconstrained subproblem solutions converge from outside the feasible region. 

When a system is constrained, the factor space is divided into feasible regions and nonfeasible regions. A feasible region contains permissible or desirable combinations of factor levels, or gives an acceptable response. A nonfeasible region contains prohibited or undesirable combinations of factor levels, or gives an unacceptable response. 

To obtain a meaningful extremum problem the number of experiments and the set of feasible vectors of the independent variables T are fixed. In most cases T is defined by inequalities x1- < x < x, i = l,2,...,k. Though introducing penalty functions such constrained extremum problems can be solved by the methods and modules described in Section 2.4, this direct approach is usually very inefficient. In fact, experiment design is not easy. The dimensionality of the extremum problem is high, the extrema are partly on the boundaries of the feasible region T, and since the objective functions are... 

FIGURE 8.10 Constrained region in three-component mixtures, (a) The feasible region (bold lines) shaped as inverted simplex lies entirely within the original simplex, (b) Part of the inverted simplex lies outside the feasible region (bold lines) and has irregular shape. 

The concept of convexity is fundamental in optimization. Many practical problems possess this property, which generally makes them easier to solve both in theory and practice. If the objective function in the optimization problem (1) and the feasible region are both convex, then any local solution of the problem is in fact a global solution. The term convex programming is used to describe a special case of the general constrained optimization problem in which ... 

The Data Collaboration approach casts problems as constrained optimization over the feasible region, drawn on the entire knowledge... 

Start at the pinch. The pinch is the most constrained region of the problem. At the pinch, A Tmin exists between all hot and cold streams. As a result, the number of feasible matches in this region is severely restricted. Quite often there... 

To a first approximation, the composition of the distillate and bottoms of a single-feed continuous distillation column lies on the same residue curve. Therefore, for systems having separatrices and multiple regions, distillation composition profiles are also constrained to lie in specific regions. The precise boundaries of these distillation regions are a function of reflux ratio, but they are closely approximated by the RCM separatrices. If a separatrix exists in a system, a corresponding distillation boundary also exists. Also, mass balance constraints require that the distillate composition, the bottoms composition, and the net feed composition plotted on an RCM for any feasible distillation be collinear and spaced in relation to distillate and bottoms flows according to the well-known lever rule. 

Grove and Harrison (1999) investigated the feasibility of obtaining Th-Pb age profiles in the surface regions of natural monazites and found that ion intensities were adequate to resolve age differences of <1 Myr with better than 500 A depth resolution in late Tertiary monazites. These age gradients were then used to extract continuous thermal history information from which they constrained the displacement history of a Himalayan thrust. The sputtering of natural surfaces was found to yield inter-element calibration plots of similar reproducibility to that of polished surfaces. 

Table X shows some typical optimization results obtained when constraining specific sensory characteristics of each mixture to lie within specified boundaries. Not all constraints work, however. The chemist, perfumer or fragrance developer must be sure that the constraints are compatible with the mixture. It does little good to constrain the sensory characteristics to lie in a region that is never reached by any feasible mixture of the odorants.

Two solutions are shown in Figure 18.3. The first is for the unconstrained case, where the entire region is feasible. The solution, shown in Figure 18.3a, is at. 2 = 0 and x, = 0, where y = 0. The second is a constrained case with the inequality constraint ... 

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. 

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