Introduction to Solving Linear Programs using LINDO V Table of LINDO commands VI The Simplex Algorithm VII Integer and Integer-Linear Programming: A Branch-and-Bound Algorithm VIII Selected Bibliography Deterministic Optimization and Design Jay R. Lund UC Davis Winter 2000 92 A Derivation of the Lagrange Multiplier Method Problem: Max Z = f(X 1 , X 2 ) Subject To (S.T.): g(X 1 , X 2 ) = b, b is a constant If constraint could be reformulated as:X 2 =g 1 X 1 , then the constraint could be substituted into f X 1 ,X 2 and solved as unconstrained optimization. (This is often not possible or expedient. First-order Conditions (2 of 'em) To Max Z, subject to the constraint, a first-order condition is: (1) df dX 1 = 0 = #f #X 1 +<F15.6

Just as modem general-purpose programming languages (e.g. C--, Java) are supported by a suite of tools (debuggers, profilers, etc.), mathematical programming languages need supporting tools. MProbe is an example of a suite of tools supporting a mathematical programming language, in this case AMPL. MProbe includes tools for empirically estimating the shape of nonlinear functions of many variables, nonlinearly-constrained region shape, the effect of the objective shape on the ability to find a global optimum, tools for estimating the effectiveness of constraints and for navigating through the model, among others.