An Augmented Lagrangian Method for Cardinality-Constrained Optimization Problems

A reformulation of cardinality-constrained optimization problems into continuous nonlinear optimization problems with an orthogonality-type constraint has gained some popularity during the last few years. Due to the special structure of the constraints, the reformulation violates many standard assumptions and therefore is often solved using specialized algorithms. In contrast to this, we investigate the viability of using a standard safeguarded multiplier penalty method without any problem-tailored modifications to solve the reformulated problem. We prove global convergence towards an (essentially strongly) stationary point under a suitable problem-tailored quasinormality constraint qualification. Numerical experiments illustrating the performance of the method in comparison to regularization-based approaches are provided.

Convex duality for principal frequencies

<abstract><p>We consider the sharp Sobolev-Poincaré constant for the embedding of $ W^{1, 2}_0(\Omega) $ into $ L^q(\Omega) $. We show that such a constant exhibits an unexpected dual variational formulation, in the range $ 1 &lt; q &lt; 2 $. Namely, this can be written as a convex minimization problem, under a divergence–type constraint. This is particularly useful in order to prove lower bounds. The result generalizes what happens for the torsional rigidity (corresponding to $ q = 1 $) and extends up to the case of the first eigenvalue of the Dirichlet-Laplacian (i.e., to $ q = 2 $).</p></abstract>

Collapse Behavior of Tensegrity Barrel-Vault Structures Based on Di-Pyramid (DP) Units

In this study, the collapse behavior of a family of tensegrity structures, i.e. di-pyramid (DP) barrel-vaults that can offer promising solutions for civil engineering applications is analyzed. Depending on whether struts’ snap or cables’ rupture dictate the occurrence of overall collapse, two different designs are considered. The effects of geometric parameters, self-stress properties, loading type, boundary conditions and strengthening schemes on the structural behavior are discussed. It is found that the structures with symmetric and ridge loading types undergo bifurcation type instability instead of limit point which is encountered in the cases with asymmetric loading type. Constraint against lateral thrust is beneficial in improving strength and initial stiffness of the studied cases, by as much as 60% and 90%, respectively. In most cases, the rate of strength variation associated with increasing self-stress levels is quite slow, while the slackness load increases by at least 400% being the primary achievements. By using non-uniform self-stress distribution, the initial stiffness of these structures can be increased up to 240%. Increasing the rise-to-span ratio improves the initial stiffness and collapse strength of the structure significantly at the expense of expedition of cables slackness. Significant gains in collapse resistance of these structures under symmetric loading are obtained with strengthened critical struts or cables, depending on which collapse case dominates, but the initial stiffness is generally not influenced by these schemes.

Asymptotic stationarity and regularity for nonsmooth optimization problems

Based on the tools of limiting variational analysis, we derive a sequential necessary optimality condition for nonsmooth mathematical programs which holds without any additional assumptions. In order to ensure that stationary points in this new sense are already Mordukhovich-stationary, the presence of a constraint qualification which we call AM-regularity is necessary. We investigate the relationship between AM-regularity and other constraint qualifications from nonsmooth optimization like metric (sub-)regularity of the underlying feasibility mapping. Our findings are applied to optimization problems with geometric and, particularly, disjunctive constraints. This way, it is shown that AM-regularity recovers recently introduced cone-continuity-type constraint qualifications, sometimes referred to as AKKT-regularity, from standard nonlinear and complementarity-constrained optimization. Finally, we discuss some consequences of AM-regularity for the limiting variational calculus.