A Nonlinear Lagrangian Approach to Constrained Optimization Problems

Abstract

In this paper we study nonlinear Lagrangian functions for constrained optimization problems which are, in general, nonlinear with respect to the objective function. We establish an equivalence between two types of zero duality gap properties, which are described using augmented Lagrangian dual functions and nonlinear Lagrangian dual functions, respectively. Furthermore, we show the existence of a path of optimal solutions generated by nonlinear Lagrangian problems and show its convergence toward the optimal set of the original problem. We analyze the convergence of several classes of nonlinear Lagrangian problems in terms of their first and second order necessary optimality conditions.

Dynamical systems and topology optimization

“…By letting β = λ 3 ∆t and by doing other appropriate identifications we obtain the iteration formula (25). Remark: By doing a slightly different time discretization, (25) can be derived from (26) without the need to approximate the exponent as above: We rewrite the quotient of the left hand side of (26) as follows:…”

“…For constrained optimization problems, which are obviously the ones of relevance for structural optimization, there are different possibilities for rewriting these as dynamical systems, see, e.g., [20,26,27]. In the following we will use a dynamical system based on local projections, see [17].…”

Semismoothness of Spectral Functions

“…We note that the LC 1 problem is also known as C 1,1 data in [9], where second-order analysis of the underlying function is conducted. For further development along this line, see [30,31] and the references therein.…”

“…Another interesting case is when f is an SC 1 function, i.e., f is not only an LC 1 function, but also its derivative function is semismooth. For both cases, we will show that (f • λ) is an LC and SC 1 functions is that they constitute a class of minimization problems which can be solved by Newton-type methods (see [6,20,22]) and by penalty-type methods (see [31,30]). …”

Computing minimum distance between two implicit algebraic surfaces

“…the gradient flow method [4,8,38,43]. To compute all roots in a given area of interest, several global solution methods such as algebraic and hybrid methods, homotopy methods and subdivision methods [31] are useful.…”