Lower semi-continuity for $\mathcal A$-quasiconvex functionals under convex restrictions

We show weak lower semi-continuity of functionals assuming the new notion of a ``convexly constrained'' $\mathcal A$-quasiconvex integrand. We assume $\mathcal A$-quasiconvexity only for functions defined on a set $K$ which is convex. Assuming this and sufficient integrability of the sequence we show that the functional is still (sequentially) weakly lower semi-continuous along weakly convergent ``convexly constrained'' $\mathcal A$-free sequences. In a motivating example, the integrand is $-\det^{\frac{1}{d-1}}$ and the convex constraint is positive semi-definiteness of a matrix field.

FR-type algorithm for finding approximate solutions to nonlinear monotone operator equations

This paper focuses on the problem of convex constraint nonlinear equations involving monotone operators in Euclidean space. A Fletcher and Reeves type derivative-free conjugate gradient method is proposed. The proposed method is designed to ensure the descent property of the search direction at each iteration. Furthermore, the convergence of the proposed method is proved under the assumption that the underlying operator is monotone and Lipschitz continuous. The numerical results show that the method is efficient for the given test problems.

Modified Hager–Zhang conjugate gradient methods via singular value analysis for solving monotone nonlinear equations with convex constraint

Following a recent attempt by Waziri et al. [2019] to find an appropriate choice for the nonnegative parameter of the Hager–Zhang conjugate gradient method, we have proposed two adaptive options for the Hager–Zhang nonnegative parameter by analyzing the search direction matrix. We also used the proposed parameters with the projection technique to solve convex constraint monotone equations. Furthermore, the global convergence of the methods is proved using some proper assumptions. Finally, the efficacy of the proposed methods is demonstrated using a number of numerical examples.

Regularity conditions and Farkas-type results for systems with fractional functions

This paper deals with some new versions of Farkas-type results for a system involving cone convex constraint, a geometrical constraint as well as a fractional function. We first introduce some new notions of regularity conditions in terms of the epigraphs of the conjugate functions. By using these regularity conditions, we obtain some new Farkas-type results for this system using an approach based on the theory of conjugate duality for convex or DC optimization problems. Moreover, we also show that some recently obtained results in the literature can be rediscovered as special cases of our main results.

Gapsplit: efficient random sampling for non-convex constraint-based models

Summary Gapsplit generates random samples from convex and non-convex constraint-based models by targeting under-sampled regions of the solution space. Gapsplit provides uniform coverage of linear, mixed-integer and general non-linear models. Availability and implementation Python and Matlab source code are freely available at http://jensenlab.net/tools. Supplementary information Supplementary data are available at Bioinformatics online.