A separate set of constraints is needed for every variable in the signomial term. i These constraints are then approximated with the piecewise linear expression given above. The nonlinear transformation constraints for the signomials are first rewritten as (ET): X i = ln ( x i ) (concave) (IT): X i = x 1 i (convex) (PT): X i = x i R, X i = x 1 R, R > 1 (convex). A piecewise linear function will underestimate a concave function and overestimate a convex function. Any piecewise linear formulation will work fine and SOS 2 variables could preferably be used to expedite the calculations (see for instance). Note that the formulation above is not optimal for performance, but is used for illustration purpose only. A continuous one-dimensional function f ( x ) can be approximated on a closed interval with a piecewise linear function which coincides with f ( x ) at least at some given break points. the latter formulation the binary variables i can be omitted, since the logic is handled by the solver using SOS rules. The up- dated MINLP has four additional variables, two binary and two continuous, and two additional inequality. The NLP-heuristic includes the point (7/6, 0.5) in the grid and alternative 3 (midpoint) the point (2.25, 1.25). If updating alternative 1 is used the point (1.1143, 0.3429) is added to the grid and the problem is resolved. Once again we return to Example 1 and the ET approach. This follows from the compactness and continuity assumption of problem (P). Convergence for the method can generally be ensured by periodically using updating alternative 3. If updating alternative 1 is used the grids are updated for all variables that do not lie exactly at an existing grid point at the solution to the convexified problem. the partition is done in several dimensions.
In this method several new grid points are added in each iteration, i.e.
In classical branch and bound methods for global optimization the partition of the space is done for only one variable at a time. obtain a more detailed proof of convergence and the convergence properties of the method, the reader is referred to.
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