Nonlinear Programming--Sequential Unconstrained Minimization Techniques

Abstract

inequality constraints defined so that no penalty is imposed if an inequality is satisfied-i.e., h 0 if g} (X) < 0 1 if g¡ (X) > 0 , j-A; + 1,-f-2,. .. ,m The a/s are positive scaling factors which weight the constraints relative to each other to facilitate the hillclimbing procedure. A method for choosing these factors automatically in the course of the optimization is described by Keefer and Gottfried (1970). This strategy, unlike many optimization procedures, does NPTP = mapping array for pattern variables onto set of independent variables NPTS = mapping array for simplex variables onto set of independent variables NVS = number of simplex variables NVSOLD = previous number of simplex variables NVP = number of pattern variables X = vector of independent variables xt = tth independent variable y = objective function z = modified objective function Greek Letters not require a starting point (initial value for X) that satisfies the constraints. This can be important in certain complex applications wherein finding a "feasible point" is the most difficult part of the problem. In such cases the effort required to obtain the final solution by this strategy is often not much greater than that required to find the starting point for some other optimization procedures. Furthermore, once a feasible point has been attained, this strategy does not limit subsequent moves by the optimizer to the feasible region. This can be a desirable feature in problems where the optimizer would have a difficult path to follow to reach the optimum if it were confined to the feasible region. This paper is the second in a study of the chemical reactions encountered in the Engel-Precht process of K2CO3 manufacture. The first reaction investigated was the hydration of MgO to Mg(OH)2 (Smithson and Bakhshi, 1969). The carbonation of MgO is of interest because it is one method of preparing MgC03-3H20 which in turn is used to produce MgC03· KHC03-4H20 (Engel salt) in the Engel-Precht process. In previous studies (Evans and St. Claire, 1949; Horiguchi, 1954) it was observed that MgO reacts with C02 and water 1 To whom correspondence should be addressed. to form a solution of Mg(HC03)2 which subsequently decomposes , precipitating magnesium carbonate (tri-or penta-hydrate depending on the temperature). MgO + 2COa + H20-^ Mg(HCOa)2 aqueous (1) >14°C Mg(HC03)2 + 2H20-MgC03 · 3H20 + C02 (2) <14°C Mg(HC03)2 + 4H20-MgCOa · 5H20 + C02 (3) The time and rate at which the precipitation occurs varied with the initial concentration of MgO, C02 flow rate, stirring