Relaxation methods with step regulation for solving constrained optimization problems
We study the problem \[ \min_{x\in D} f(x),\tag{1} \] where \(f(x)\) is a continuously differentiable pseudoconvex function satisfying Condition A (introduced in [\textit{Z. R. Gabidulinna}, J. Sov. Math. 50, No. 5, 1803-1809 (1990); translation from Issled. Prikl. Mat. 14, 15-25 (1987; Zbl 0706.90061)]): A continuous function \(f\) satisfies Condition A on the set \(D\) if there exists \(\kappa>0\) such that \[ f(\alpha x+(1- \alpha)y)\geq \alpha f(x)+ (1-\alpha) f(y)- \alpha(1- \alpha)\kappa\| x-y\|^2\;\forall x,y\in D,\;\forall\alpha\in [0,1]. \] We propose a relaxation method for solving the problem (1). Depending on the method of choosing the size of the recursion step, one can obtain various versions of the algorithm.
continuously differentiable pseudoconvex function
scientific article; zbMATH DE number 894497