Optimization minimization/maximization problem

Stochastic optimization methods described previously, such as simulated annealing, can also be used to solve the general nonlinear programming problem. These have the advantage that the search is sometimes allowed to move uphill in a minimization problem, rather than always searching for a downhill move. Or, in a maximization problem, the search is sometimes allowed to move downhill, rather than always searching for an uphill move. In this way, the technique is less vulnerable to the problems associated with local optima. 

The addition of inequality constraints complicates the optimization. These inequality constraints can form convex or nonconvex regions. If the region is nonconvex, then this means that the search can be attracted to a local optimum, even if the objective function is convex in the case of a minimization problem or concave in the case of a maximization problem. In the case that a set of inequality constraints is linear, the resulting region is always convex. 

For more the details about the MPC CB software and the operations conditions, the reader can refer to [12]. The optimal minimization of the drying time rmder constraints may be equivalent to define the performance index as the maximization of the velocity of the sublimation interface. Since MPC CB solves a rninirnization problem, the objective function is ... 

Optimal process synthesis Two problems (a) chemical process optimization for maximization of net present value (NPV) while minimizing uncertainty in the future demand of two products, and (b) utility system optimization for minimization of both total annual cost and CO2 emission. Multi-Criteria Branch and Bound (MCBB) Algorithm The existing MCBB algorithm was modified to increase speed, reliability and suitahility for a wide range of applications. Mavrotas and Diakoulaki (2005)... 

In a pair-wise comparison of two solutions within the Pareto domain, a rule will contain at least one zero and, as a result, a minimum of one objective function will always be sacrificed. RSM cannot be used for a two-objective optimization because the decision-maker will have to make a clear choice between one of the two objective functions and the preference rule can only be (10) or (01). For instance, choosing (01) automatically means that the optimal solution is the lowest possible value of the second objective in the case of a minimization problem, and the highest possible value for a maximization problem. RSM reduces a two-objective problem to a SOO problem. As the number of objectives increases, the overall effect of losing at least one dimension in objective space diminishes significantly. 

When dealing with difficult discrete optimization problems, it is natural to search for related, but easier optimization models that can aid in the analysis. Relaxations are auxiliary optimization problems of this sort formed by weakening either the constraints or the objective function of the main problem. Specifically, an optimization problem P) is said to be a constraint relaxation of another optimization problem (P) if every solution feasible to (P) is also feasible for (P). Similarly, maximize problem (respectively minimize problem) P) is an objective relaxation of another maximize (respectively minimize) problem (P) if the two problems have the same feasible solutions and the objective function value in P) of any feasible solution is (respectively <) the objective function value of the same solution in (P). 

The objective function value of an optimal solution to a relaxation bounds the optimed objective function value of the main problem. Specifically, relaxation optima provide lower bounds for minimize problems and upper bounds for maximize problems. 

A desirable feature of branch and bound is that a bound on the value of a global optimal solution is always available, so that the algorithm need not be run to termination in order to bound the error of accepting the incumbent solution as approximately optimal. The best stored bound of CAND (least for minimize problems, highest for maximize problems) always provides such a bound. Thus, for example, after node 8 is processed in the example of Figure 2, it is certain that any solution to the full ILF) will cost at least min 12.6,13.4 = 12.6. Stopping at that point with the incumbent solution of value 14 would produce at most (14 - 12.6)/12.6 = 11.1% error. 

We consider a minimization rather than a maximization problem for the sake of notational convenience.) Here C R is a set of permissible values of the vector x of decision variables and is referred to as the feasible set of problem (11). Often x is defined by a (finite) number of smooth (or even linear) constraints. In some other situations the set x is finite. In that case problem (11) is called a discrete stochastic optimization problem (this should not be confused with the case of discrete probability distributions). Variable random vector, or in more involved cases as a random process. In the abstract fiamework we can view as an element of the probability space (fi, 5, P) with the known probability measure (distribution) P. 

Application 1 Maximize operating profit In Chapter 19, real-time optimization was considered problems where the operating profit was expressed in terms of product values and feedstock and utility costs. If the product, feedstock, and utility flow rates are manipulated or disturbance variables in the MPC control structure, they can be included in objective function Js- In order to maximize the operating profit OP), the objective function is specified to be Js = —OP, because minimizing is equivalent to maximizing The weighting matrices for two quadratic terms, Qsp and Rsp, are set equal to zero. 

Finding the best solution when a large number of variables are involved is a fundamental engineering activity. The optimal solution is with respect to some critical resource, most often the cost (or profit) measured in doUars. For some problems, the optimum may be defined as, eg, minimum solvent recovery. The calculated variable that is maximized or minimized is called the objective or the objective function. 

Formulation of the Objective Function The formulation of objective functions is one of the crucial steps in the application of optimization to a practical problem. You must be able to translate the desired objective into mathematical terms. In the chemical process industries, the obective function often is expressed in units of currency (e.g., U.S. dollars) because the normal industrial goal is to minimize costs or maximize profits subject to a variety of constraints. 

Maximization, Minimization, and Minmax.—We shall discuss three wide classes of optimization problems that illustrate the ideas of the title of this section as used in operations research. They are ... 

Optimization problems in process design are usually concerned with maximizing or minimizing an objective function. The objective function might typically be to maximize economic potential or minimize cost. For example, consider the recovery of heat from a hot waste stream. A heat exchanger could be installed to recover the waste heat. The heat recovery is illustrated in Figure 3.1a as a plot of temperature versus enthalpy. There is heat available in the hot stream to be recovered to preheat the cold stream. But how much heat should be recovered Expressions can be written for the recovered heat as ... 

All of these variables must be varied in order to minimize the total cost or maximize the economic potential (see Chapter 2). This is a complex optimization problem involving both continuous variables (e.g. batch size) and integer variables (e.g. number of units in parallel) and is outside the scope of the present text9. 

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