By M. Avriel (auth.), Mordecai Avriel (eds.)

In 1961, C. Zener, then Director of technology at Westinghouse Corpora tion, and a member of the U. S. nationwide Academy of Sciences who has made vital contributions to physics and engineering, released a quick article within the complaints of the nationwide Academy of Sciences entitled" A Mathe matical relief in Optimizing Engineering layout. " listed here Zener thought of the matter of discovering an optimum engineering layout which could frequently be expressed because the challenge of minimizing a numerical price functionality, termed a "generalized polynomial," which includes a sum of phrases, the place each one time period is a fabricated from a favorable consistent and the layout variables, raised to arbitrary powers. He saw that if the variety of phrases exceeds the variety of variables by means of one, the optimum values of the layout variables will be simply came upon through fixing a suite of linear equations. moreover, convinced invariances of the relative contribution of every time period to the complete expense might be deduced. The mathematical intricacies in Zener's technique quickly raised the interest of R. J. Duffin, the celebrated mathematician from Carnegie Mellon college who joined forces with Zener in laying the rigorous mathematical foundations of optimizing generalized polynomials. Interes tingly, the research of optimality stipulations and homes of the optimum strategies in such difficulties have been conducted via Duffin and Zener due to inequalities, instead of the extra universal procedure of the Kuhn-Tucker theory.

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Actually, signomial optimization problems (as well as more general "algebraic optimization problems") should usually be reduced to much simpler signomial optimization problems prior to their transformation to an appropriate Problem A. To see how to reduce such problems to signomial optimization problems in which each signomial has at most two terms, both of which have the same sign, consult Duffin (Ref. 25) and Duffin and Peterson (Refs. 13 and 15). If such problems could be further reduced to signomial problems in which each signomial has only a single term, all algebraic optimization problems (and hence essentially all optimization problems involving only continuous functions) could be reduced to (finitedimensional) linear programming problems.

S), I k~l Vkakj = f i~l YiCij. :. o} . 5. Hdx2k' X2k+l) Let = [(X2k + d 2d(X2k+l + d2k +1)]1/2 (k=0,1, ... 10 and Elementary Formula (b) in )_{-d2kY2k-d2k+lY2k+l H *( k Y2k, Y2k+l -00 (x+d~O). 2, if (Y2kY2k+l)I/2 ~ 1/2, Y> 0, otherwise, and we have the following dual pair: max {(xo + dO)(XI + d 1) I(X2k + d 2k )(X2k+l + d 2k +1) ~ b~ (k = 1, 2, ... Ab Y > 0, Y E~} = min r~:1 diYi -2 kt bk(Y2kY2k+l)I/21 YOYI ~t Y> 0, YE ~}. If I 9» = {X Xi = jt (20) aijZj (i = 0, 1, ... ,2m + 1) }, then (19), (20) can be written as max {(L aOjzj + do)(L aljZj + d 1) I(L a2k,jZj + d 2k )(L a2k+l,jZj + d 2k+1) ~ b~}, minr~:l diYi-2 k~l bk(Y2kY2k+l)1/2IYOYl~tY>0, 2~:1 Yiaij=O}.

We suppose that for each "stage" i there is a cost 9i(/' d i ) that depends on the ith "state" ~' and the ith "decision" d i , where the domain of the cost function 9. is the Cartesian product Ri x Di of the ith "state set" R, and the ith "decision set" D i • We also suppose that the "initial state" -t i is determined by the "initial decision" d l through the equation -t i = Bid l and that each subsequent state ~' is determined by both the ith decision d i and the (i -l)th state -t i - I through the "transition equation" ~i = A3 i - 1 + Bid', where Ai and B, are constant matrices.