Sqp software

The convergence properties of the basic sequential QP algorithm can be improved by using a line search. Often these two aims conflict, so it is necessary to weigh their relative importance and define a merit or penalty function, which we can use as a criterion for determining whether or not one point is better than another.

Skip to main content. In practice, however, it is likely that the divergence will not be an invertible matrix because variables are likely to be linearly bound from above and below.

The improvement direction "p" for the Newton's Method iterations is thus typically found in a more indirect fashion: with a quadratic minimization sub-problem that is solved using quadratic algorithms.

The subproblem is derived as follows: Since p is an incremental change to the objective function, this equation then resembles a two-term Taylor Series for the derivative of the objective function, which shows that a Taylor expansion with the increment p as a variable is equivalent to a Newton iteration. Decomposing the different equations within this system and cutting the second order term in half to match Taylor Series concepts, a minimization sub-problem can be obtained.

This problem is quadratic and thus must be solved with non-linear methods, which once again introduces the need to solve a non-linear problem into the algorithm, but this predictable sub-problem with one variable is much easier to tackle than the parent problem.

This example problem was chosen for being highly non linear but also easy to solve by inspection as a reference. The objective function Z is a trigonometric identity: The first constraint then just restricts the feasible zone to the first half of a period of the sine function, making the problem convex. The maximum of the sine function within this region occurs at , as shown in Figure 1.

The last constraint then makes the problem easy to solve algebraically: and Now, the problem will be solved using the sequential quadratic programming algorithm. The Lagrangian funtion with its gradient and divergence are as follows:. The first important limitation in using SQP is now apparent: the divergence matrix is not invertible because it is not full rank.

We will switch to the quadratic minimization sub-problem above, but even with this alternate framework, the gradient of the constraints must be full rank. This can be handled for now by artificially constraining the problem a bit further so that the derivatives of the inequality constraints are not linearly dependent. This can be accomplished through a small modification to the constraint.

If , then it is also true that The addition to the left-hand-side is relatively close to zero for the range of possible values of , so the feasible region has not changed much. This complication is certainly annoying, however, for a problem that's easily solved by inspection.

It illustrates that SQP is truly best for problems with highly non-linear objectives and constraints. Discover the five key actions you can take to help create a future with zero emissions, zero waste, and zero inequality. Work from home, car sharing, omnichannel buying, resilient supply chains, e-bikes, smart meters, personalized on-demand experiences — all thanks to technology and the human spirit.

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