Knowledge that Transforms

Disaggregate demand models predict the choice behavior of individual consumers. But while such models predict choice probabilities (0 < p < 1), they must be tested against (0, 1) choice behavior. This paper uses information theory to derive three complementary tests that help analysts select a “best” disaggregate model. “Usefulness” measures the percentage of uncertainty (entropy) explained by the information the model provides. It provides theoretic rigor and intuitive appeal to the commonly used likelihood ratio index and leads to important practical extensions. “Accuracy” is a new two-tailed normal test that determines whether the (0, 1) observations are reasonable under the hypothesis that the model is valid. “Significance” is the standard chi-squared test to determine whether a null model can be rejected. This paper also extends the information test to examine the relationships among successively more powerful null hypotheses. For example, in a logit model one can quantify (1) the contribution due to knowing aggregate market shares, (2) the incremental contribution due to knowing choice set restrictions, and (3) the final incremental contribution due to the explanatory variables. Further extensions provide “explanable uncertainty” measures applicable if choice frequencies are observed. Market research and transportation analysis empirical examples are given.

Inner approximation algorithms have had two major roles in the mathematical programming literature. Their first role was in the construction of algorithms for the decomposition of large-scale mathematical programs, such as in the Dantzig-Wolfe decomposition principle. However, recently they have been used in the creation of algorithms that locate Kuhn-Tucker solutions to nonconvex programs. Avriel and Williams' (Avriel, M., A. C. Williams. 1970. Complementary geometric programming. SIAM J. Appl. Math. 19 125–141.) complementary geometric programming algorithm, Duffin and Peterson's (Duffin, R. J., E. L. Peterson. 1972. Reversed geometric programs treated by harmonic means. Indiana Univ. Math. J. 22 531–550.) reversed geometric programming algorithms, Reklaitis and Wilde’s (Reklaitis, G. V., D. J. Wilde. 1974. Geometric programming via a primal auxiliary problem. AIIE Trans. 6 308–317.) primal reversed geometric programming algorithm, and Bitran and Novaes' (Bitran, G. R., A. G. Novaes. 1973. Linear programming with a fractional objective function. Opns. Res. 21 22–29.) linear fractional programming algorithm are all examples of this class of inner approximation algorithms. A sequence of approximating convex programs are solved in each of these algorithms. Rosen's (Rosen, J. B. 1966. Iterative solution of nonlinear optimal control problems. SIAM J. Control 4 223–244.) inner approximation algorithm is a special case of the general inner approximation algorithm presented in this note.

This paper gives general conditions under which a collection of optimization problems, with the objective function and the constraint set depending on a parameter, has optimal solutions that are an isotone function of the parameter. Relating to this, we present a theory that explores and elaborates on the problem of minimizing a submodular function on a lattice.