Part 4 Nonlinear Programming 4.3 Successive Linear Programming

Approach 3 Cutting Plane Method

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

Basic Strategy

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

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.

Basic Ideas of Kelley’s Algorithm

Nonlinear Objective Function

Generation of Cuts

Generation of Cuts

Generation of Cuts Kelley proposed that : Only the linearization of the most violated constraint be used to construct a cut.

Kelley’s Algorithm

Kelley’s Algorithm – Step 1

Kelley’s Algorithm – Step 2

Kelley’s Algorithm – Step 3

Kelley’s Algorithm – Step 4

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.

Requirement of Convexity

Remark

Cut-Deletion Procedure

Step 4a

Step 4b