PART II

FRAMEWORK

EQUATIONS play a dominant role in structuring problems. An equation is a statement of identity or a conditional equality (holding only for certain values of the unknowns) between two quantities. Equations usually involve two types of quantities: parameters and variables. The parameters are known or controlled numerical values which enter into the formulation. The variables are unknown and are allowed to assume any value in a specified range. The problem is to determine these unknowns. A system of equations may be underdetermined if it has more variables than the number of equations and overdetermined in the opposite case. In the former case, some of the variables are arbitrary and the remaining variables may be expressed as functions of the arbitrary variables. Ordinarily, if the system of equations is not redundant and the number of variables is equal to the number of equations, it is possible to obtain specific values for the variables as solutions. The values of the variables depend on those of the coefficients, and solutions of equations are thus functions of their parameters.

An optimization problem has a structure involving a set of constraints given in the form of equations or inequalities and a function, frequently called an objective function to be maximized or minimized subject to the constraints. Optimization is a normative or prescriptive field in which among all possible solutions which satisfy the system of constraints, that which yields a maximum or minimum (called an optimum) to the objective function is sought. Often, the process of solving an optimization problem can be reduced to a process of solving a set of equations.

Stochastic processes can be visualized formally in terms of probability theory and equations. Even when the structure of a problem is probabilistic, we use equations and inequalities for its representation. In this case the variables of the problem are subject to the laws of probability, and the method of solution differs from that of a nonprobabilistic problem; the parameters of the equations may also be subject to randomness, adding to the complexity of the solution. Optimization problems in which probability plays a central role have come to the fore in recent years; such formulations are thought to be closer to reality and to give more faithful representations than non-probabilistic equations. So, the three pillars—equations, optimization, and probability—can be seen to be closely interdependent in the process of modeling.