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The purpose of this article is to introduce a gradient-flow algorithm for solving generic equality or inequality constrained optimization problems, which is suited for shape optimization applications. We rely on a variant of the Ordinary Differential Equation (ODE) approach proposed by Yamashita for equality constrained problems: the search direction is a combination of a null space step and a range space step, which are aimed to reduce the value of the minimized objective function and the violation of the constraints, respectively. Our first contribution is to propose an extension of this ODE approach to optimization problems featuring both equality and inequality constraints. In the literature, a common practice consists in reducing inequality constraints to equality constraints by the introduction of additional slack variables. Here, we rather solve their local combinatorial character by computing the projection of the gradient of the objective function onto the cone of feasible directions. This is achieved by solving a dual quadratic programming subproblem whose size equals the number of active or violated constraints, and which allows to identify the inequality constraints which should remain tangent to the optimization trajectory. Our second contribution is a formulation of our gradient flow in the context of-infinite-dimensional-Hilbert space settings. This allows to extend the method to quite general optimization sets equipped with a suitable manifold structure, and notably to sets of shapes as it occurs in shape optimization with the framework of Hadamard's boundary variation method. The cornerstone of this latter setting is the classical operation of extension and regularization of shape derivatives. Some numerical comparisons on simple academic examples are performed to illustrate the behavior of our algorithm. Its numerical efficiency and ease of implementation are finally demonstrated on more realistic shape optimization problems.

Null space gradient flows for constrained optimization with applications to shape optimization