A New Conical Regularization for Some Optimization and Optimal Control Problems: Convergence Analysis and Finite Element Discretization

On Henig Regularization of Material Design Problems for Quasi-Linear <i>p</i>-Biharmonic Equation

“…As was shown in the recent publication [7], the proposed approach is numerically viable for state-constrained optimal control problems with the state equation given by linear partial differential equations. In particular, using the finite element discretization of the Henig dilating cone of positive functions, it has been shown in [7] that the above approximation scheme, called conical regularization, where the regularization is done by replacing the ordering cone with a family of dilating cones, leads to a finite-dimensional optimization problem which can conveniently be treated by known numerical techniques.…”

“…In particular, using the finite element discretization of the Henig dilating cone of positive functions, it has been shown in [7] that the above approximation scheme, called conical regularization, where the regularization is done by replacing the ordering cone with a family of dilating cones, leads to a finite-dimensional optimization problem which can conveniently be treated by known numerical techniques. The non-emptiness of the feasible set for the stateconstrained OCPs is an open question even for the simplest situation.…”

A Henig conical regularization approach for circumventing the Slater conundrum in linearly $\ell_{+}^{p}$-constrained least squares problems

“…which provides a non-trivial feasible point such that the regularized optimality system is solvable (see [15]). Although the original regularization scheme given in [15, Theorems 3.1-3] is given for a Hilbert constraint space Y , the proofs of that results are still valid for a general Banach space.…”

“…This problem, with a possible singular multiplier in p = ∞, and non-existence of interior points in the case p ∈ [1, ∞), has recently been named as the Slater conundrum in the literature, [4,20]. In this context, in this paper, we exploit a general technique, dubbed the conical regularization, which was devised in [15] to circumvent the difficulties associated with the failure of the Slate-type constraint qualifications. For (Q p ), the conical regularization consists of replacing the ordering cone by a family of dilating cones.…”

“…For (Q p ), the conical regularization consists of replacing the ordering cone by a family of dilating cones. Conical regularization scheme has been tested by using different families of dilating cones [10,11,12], it has been applied to different PDE constrained optimization problems, see [14,15,18], and extended to multiobjective and vector optimization problems, see [13,25].…”