Optimization with equality and inequality constraints using parameter continuation

Abstract

We generalize the successive continuation paradigm introduced by Kernévez and Doedel [16] for locating locally optimal solutions of constrained optimization problems to the case of simultaneous equality and inequality constraints. The analysis shows that potential optima may be found at the end of a sequence of easily-initialized separate stages of continuation, without the need to seed the first stage of continuation with nonzero values for the corresponding Lagrange multipliers. A key enabler of the proposed generalization is the use of complementarity functions to define relaxed complementary conditions, followed by the use of continuation to arrive at the limit required by the Karush-Kuhn-Tucker theory. As a result, a successful search for optima is found to be possible also from an infeasible initial solution guess. The discussion shows that the proposed paradigm is compatible with the staged construction approach of the coco software package. This is evidenced by a modified form of the coco core used to produce the numerical results reported here. These illustrate the efficacy of the continuation approach in locating stationary solutions of an objective function along families of two-point boundary value problems and in optimal control problems.

A Unified Analytical Framework for Optimal Control Problems on Networks With Input Homogeneity

“…6. For Problem 2, the predicted stationary solution µ(t) ≡ C 2 /T requires the determination of C 2 in terms of r from the rescaled system dynamics and the coupling condition (29). We obtain similar agreement to that in Fig.…”

“…• Step 3: Successive continuation. Without a priori restriction to positively-valued µ, we use parameter continuation to satisfy the necessary conditions through a succession of separate stages [23], [24], [29], where each successive run is initialized by the solution from the previous run. Importantly, due to the linearity and homogeneity of the adjoint equations, the first run can be initialized with a solution guess with zero Lagrange multipliers.…”

“…It follows from Theorem III.2 that the candidate optimal µ with Φ h = 0 depends only on T and C 1 and is thus independent of the initial conditions z 0 and problem parameters p. By ex- tension, the optimal solution of Problem 1 is also independent of the initial conditions x i (0). In contrast, it follows from Theorems III.3 and III.4 that the candidate optimal µ depend on T , Φ, z 0 , and p due to the coupling condition (29). By extension, the optimal solutions to Problems 2 and 3 depend on the initial conditions x i (0).…”

“…Suppose that Φ h (T * , µ * ) = 0 for some T * ∈ R + and µ * ∈ C([0, T * ], R + ). If µ * and T * , additionally, satisfy the first-order necessary conditions for a stationary point of the functional (µ, T ) → T subject to the constraints G(µ) = C 1 and Φ(z(T )) = Φ given C 1 ∈ R + and Φ ∈ R, then µ * and T * satisfy the first-order necessary conditions for Problem 6 with C 2 given by one of the roots of the coupling condition (29), and vice versa.…”

“…We can apply the solution to Problem 4 directly to find the candidate optimal µ for Problem 1. In contrast, for Problems 2 and 3, we must first find C 2 from the coupling condition (29) for a given r before applying the solution to Problems 5 and 6. It is clear that the value of C 2 depends on the network connectivity.…”