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Explore the successive linear programming approach using the cutting plane method to optimize nonlinear objective functions efficiently. Understand Kelley's algorithm steps, advantages, and disadvantages for solving complex optimization problems.
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Part 4 Nonlinear Programming 4.3 Successive Linear Programming
Basic Strategy We seek to devise an algorithm that will solve this problem by solving a sequence of intermediate problems constructed by starting out with a rough approximation to the feasible region and successively improving the approximation by adding constraint estimates updated at the intermediate solution.
Basic Strategy Case (ii) gives us an indication of the possible location of the optimum. In order to improve our approximation to F in the vicinity of x^(1), we will need to modify the boundaries of Z^0 near x^(1). This can be achieved by imposing additional constraints that will exclude from Z^0 the region in the vicinity of x^(1).
Example F x2 P1 P2 x1 F(x1,x2)=-x1-x2
Example If the computations are continued in this fashion, and if with each set of cuts we can be sure that a nonempty remaining portion of Z^(0) is eliminated, then it seems reasonable that eventually a point will be reached that is feasible and that consequently will be the minimum of f(x) over F.
Generation of Cuts Kelley proposed that : Only the linearization of the most violated constraint be used to construct a cut.
Advantages • Any linearity or near linearity in the original problem is preserved and directly utilized. • The sub-problem to be solved at each major iteration is one for which the powerful techniques of LP are applicable.
Disadvantages • The algorithm generates a sequence of infeasible points. Thus, it cannot be terminated early with a “good” but perhaps not optimal point. • The size of the LP problem grows continuously. • The feasible region F has to be convex.
