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CLASSICAL THEORY OF MAXIMA AND MINIMA
- Introduction
- Analytical Methods without Constraints
- Analytical Methods Applicable for Constraints
- Necessary and Sufficient Conditions for Constrained Problems
- Closure
- References
- Problems
| Closure | top |
In the chapter we have discussed the necessary and sufficient conditions to evaluate the character of stationary points for unconstrained and constrained optimization problems. It was necessary to confine the illustration of these procedures to simple algebraic models. Even though we are not able to apply these procedures directly to the optimization of industrial processes, the concepts developed in this chapter are used many times over in the following chapters.
It is worthwhile to attempt to solve the following unconstrained economic model from the design of horizontal vapor condensers in evaporators used in water desalination plants to see one of the major limitations of the classical theory of maxima and minima. The problem is to minimize the cost given by the following equation.
C = aN-7/6 D-1 L-4/3 + b N-0.2 D0.8 L-1 + cNDL + d N-1.8 D-4.8 L (2-56)
In this equation the cost is in dollars per year; N is the number of tubes in the condenser; D is the nominal diameter of the tubes in inches; L is the tube length in feet; and a, b, c, and d are coefficients that vary with the fluids involved and the construction costs. Avriel and Wilde (18) give further details about the significance of each term. This equation is typical of the form that is obtained from assembling correlations of equipment costs and related process operating conditions for preliminary cost estimates.
Differentiating this equation with respect to the three independent variables N, D, and L, and setting the results equal to zero gives the following three equations to be solved for the values of N, D, and L that would give the minimum cost.
(-7a/6)N-13/6 D-1L-4/3 + (-0.2b)N-1.2D0.8L-1 + cDL + (-1.8d)N-2.8D-4.8L = 0
-aN-7/6 D-2 L- 4/3 + 0.8bN-0.2 D-0.2 L-1 + cNL + (-4.8d)N-1.8 D-5.8 L = 0
(-4a/3)N-7/6 D-2 L-7/5 - N-0.2 D.08 L-2 + cND + dN-1.8 D-4.8 = 0 (2-57)
There is no straightforward way to solve
this relatively complicated set of three nonlinear algebraic equations
other than numerically with a root finding procedure at this point. This
then illustrates one on the major limitations with classical methods,
i.e., if the variables in the economic model have fractional exponents,
then a set of nonlinear algebraic equations are obtained that will probably
require an iterative solution using a computer. However, as will be seen
in the next chapter on geometric programming, we will be able to obtain
the optimal solution for this economic model by solving a set of linear
algebraic equations. We will take advantage of the mathematical structure
of the problems to be able to find the optimum readily. In fact, this
will be true of most of the modern methods; they take advantage of the
mathematical structure of the optimization problem to quickly find the
best values of the independent variables and the maximum profit or minimum
cost.
| References | top |
1. Hancock, H., Theory of Maxima and Minima, Dover Publications, Inc., New York (1960).
2. Wilde, D. J. Ind. Eng. Chem., 57 (8):18 (1965).
3. Smith, C. L., R. W. Pike and P. W. Murrill, Formulation and Optimization of Mathematical Models, International Textbook Co., Scranton, Pa. (1970).
4. Wilde, D. J. and C. S. Beightler, Foundations of Optimization, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1967).
5. Wilde, D. J., Optimum Seeking Methods, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1965).
6. Beveridge, G. S. G., and R. S. Schechter, Optimization Theory and Practice, McGraw-Hill Book Co., New York (1970).
7. Cooper, Leon, Mathematical Programming for Operations Researchers and Computer Scientists, Ed. A. G. Holtzman, Marcel Dekker, Inc., New York (1981).
8. Walsh, G. R. Methods of Optimization, John Wiley and Sons, Inc., New York (1979).
9. Sivazlian, B. D. and L. E. Stanfel, Optimization Techniques in Operations Research, Prentice-Hall, Inc., Englewood Cliffs, N.J. (1975).
10. Avriel, M. Nonlinear Programming, Analysis and Methods, Prentice Hall, Inc., Englewood Cliffs, N. J. (1976).
11. Courant, R. and D. Hilbert, Methods of Mathematical Physics, vol.I, p. 164, Interscience Publishers, Inc., New York (1953).
12. Burley, D. M., Studies in Optimization, John Wiley and Sons, Inc., New York (1974).
13. Sokolnikoff, I. S., and R. M. Redheffer, Mathematics of Physics and Modern Engineering, Second Edition, McGraw - Hill Book Co., New York (1966).
14. Kuhn, H. W., and A. W. Tucker, "Nonlinear Programming," Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, Ed. Jerzy Neyman, p. 481 - 92, University of California Press, Berkeley, California (1951).
15. Bazaraa, M. S., and C. M. Shetty, Nonlinear Programming - Theory and Algorithms, John Wiley & Sons, Inc., New York (1979).
16. Gill, P. E., W. Murray, and M. H. Wright, Practical Optimization, Academic Press, New York (1981).
17. Reklaitis, G. V., A. Ravindran and K. M. Ragsdell, Engineering Optimization: Methods and Applications, John Wiley and Sons, Inc., New York (1983).
18. Avriel, M. and D. J. Wilde, "Optimal Condenser Design by Geometric Programming", I & EC Process Design and Development, Vol. 6, No. 2, p. 256 (April, 1967).
| Problems | top |
2-1. Locate the stationary points of the following functions and determine their character.
| a. y = x4/2 - x2/2 | Solution to 2-1.a |
| b. y = x7 | Solution to 2-1.b |
| c. y = x12 + x1x2 + x22 | Solution to 2-1.c |
| d. y = 2x12 + 3x22 + 4x32 - 8x1 - 12x2 - 24x3 + 110 | Solution to 2-1.d |
2-2. Find the global maximum of the function
y(x1, x2) = 5(x1 - 3)2 - 12(x2 + 5)2 + 6x1x2
in the region
0 < x1 < 10
0 < x2 < 5
2-3. Use the Jacobian determinants and obtain the two equations to be solved with the constraint equation for the following problem
Optimize: y(x1, x2, x3)
Subject to: f(x1, x2, x3) = 0
2-4. Solve the following problem by the method of constrained variation and the method of Lagrange multipliers, evaluating x1, x2 and the Lagrange multiplier l at the optimum.
Maximize: x1 + x2
Subject to: x12 + x22 = 1
2-5. Solve the following problem by the method
of Lagrange multipliers and give the character of the stationary point.
Minimize: y = 2x12 - 4x1x2 + 4x22 + x1 - 3x2
Subject to: 10x1 + 5x2 < 3
2-6. Consider the following problem
Optimize: -x12 - 2x1 + x22
Subject to: x12 + x22 - 1 < 0
a. Obtain the equation set to be solved to locate the stationary points of the above problem using the method of Lagrange multipliers. Convert the inequality constraint to an equality constraint with the slack variable x3 as x32; why?
b. Show that the following are solutions to the algebraic equations obtained in part a.
Stationary Points
A B C D
x1 -½ -½ -1 1
x2 Ö3/2 Ö-3/2 0 0
x3 0 0 0 0
l -1 -1 0 2
c. Based on the value of the function being optimized, state whether stationary points A through D are maximum, minimum, or saddle points.
2-7. The cost of operation of a continuous, stirred-tank reactor is given by the following equation.
CT = Cf cAo q + CmV
The total operating cost CT ($/hr) is the sum of the cost of the feed, Cf cAo q, and the cost of mixing, CmV. The following gives the values for the reactor.
Cf = $5.00/lb-mole of A, cost of feed
cAo = 0.04 lb-mole/ft3, initial concentration of A.
q = volumetric flow rate of feed to the reactor in ft3/hr.
Cm = $0.30/hr-ft3, cost of mixing
V = volume of reactor in ft3
We are interested in obtaining the minimum operating cost and the optimal values of the feed rate, q; reactor volume, V; and concentration in the reactor, cA. The following first order reaction takes place in the reactor
A ® B
where the rate of formation of B, rB is given by
rB= kcA
where k = 0.1 hr-1
a. If 10 lb-moles per hour of B are to be produced, give the two material balance constraint equations which restrict the values of the independent variables. (There is no B in the feed stream.)
b. Form the Lagrangian function and perform the appropriate differentiation to obtain the set of equations that would be solved for the optimal values of the independent variables. How many equations and variables are obtained?
c. Solve for the optimal values of the reactor volume, V; feed rate, q; and concentration of A in the product, CA.
2-8. Solve the following problem by the method of Lagrange multipliers, and determine the character of the stationary point.
Optimize: y = 2x12 + 2x1x2 + x22 - 20x1 - 14x2
Subject to: x1 + 3x2 < 5
2x1 - x2 < 3
2-9.(9) Solve the following problem by the method of Lagrange multipliers, and determine the character of the stationary point.
Optimize: (1/3)x1 + x2
Subject to: -x1 + x2 < 0
x1 + x2 < 3
2-10. The total feed rate to three chemical reactors in parallel is 1100 pounds per hour. Each reactor is operating with a different catalyst and conditions of temperature and pressure. The profit function for each reactor has the feed rate as the independent variable, and the parameters in the equation are determined by the catalyst and operating conditions. The profit functions for each reactor are given below.
P1 = 0.2F1 - 2(F1/100)2
P2 = 0.2F2 - 4(F2/100)2
P3 = 0.2F3 - 6(F3/100)2
Determine the maximum profit and the optimal feed rate to each reactor.
2-11. Solve the following problem by the method of Lagrange multipliers and determine the character of the stationary points.
maximize: 3x12 + 2x22
subject to: x12 + x22 < 25
9x1 - x22 < 27
2-12. Find the stationary points of the following problem, and determine their character, i.e., maximum, minimum, or saddle point.
Optimize: 2x12 + x22 - 5x1 - 4x2
Subject to: x1 + 3x2 < 5
2x1 - x2 < 4
2-13. The rate of return (ROR) is defined as the interest rate where the net present value (NPV) is zero for a specified number of years n and initial cash flow CFo. This can be formulated as an optimization problem, as follows:
Minimize: (NPV)2
For the case of constant cash flows CFj = A, develop the equation to determine the rate of return. The net present value is given by the following equation.
NPV = -|CFo| + A[1 - (i + 1)-n] / i
2-14. Derive the Lagrangian function for
n independent variables and m constraint equations, equation (2-24). Begin
by multiplying the constraint equations given in equation (2-19) by Lagrange
multipliers l1,
l2,
...,lm.
Then add all of the equations rearrange terms and obtain the result as
equation (2-24).
2-15. For sufficient conditions of the equality constraint problem to determine if the quadratic form is positive or negative definite, the signs of the roots of a polynomial can be evaluated. This characteristic polynomial is obtained by evaluating the following determinant, which includes the second partial derivatives of the Lagrangian function evaluated at the Kuhn-Tucker points, Lxj xk(x*,l) written as Ljk for simplicity, and the first partial derivative of the constraint equations evaluated at the Kuhn-Tucker point, dfj(x*)/dxk written as fjk for simplicity.
The following results are used to evaluate the type of stationary points. First, evaluate the roots of P(a) using the above equation.
If each root of P(a) is positive, then x* is a maximum. If each root of P(a) is negative, then x* is a minimum. Finally, if the roots are of mixed sign, then x* is a saddle point.
Use the results given in Example 2-10 in the above determinant, and confirm the character of the Kuhn-Tucker point by this method. (There is a comparable sufficient condition test for the unconstrained problem which is described by Sivazlian and Stanfel (9).)