Methods of ''reduced'' directions, based on a differentiable penalty function, for a problem of nonlinear programming (Q1902860)

We consider the following problem (1): \(\min\{f_0(x)|x\in \Omega\}\), \(\Omega= \{x\in E^n|f_j(x)\leq 0, j\in J\}\), where \(E^n\) is the \(n\)-dimensional Euclidean space, \(J\) is a finite set of indices, the functions \(f_j(x)\) are from \(C^{(2)}(E^n)\), \(j\in \overline J\), \(\overline J= J\cup \{0\}\). By a solution to (1) we mean a point \(x_*\) satisfying the first-order necessary conditions for a minimum: there exist numbers \(\lambda^j_*\geq 0\), \(j\in J\), such that \[ f_0'(x_*)+ \sum_{j\in J} \lambda^j_* f_j'(x_*)= 0,\quad \lambda^j_* f_j(x_*)= 0,\quad j\in J.\tag{2} \] In previous papers the authors had constructed a method for solving (1), which uses as a movement direction to so-called ``reduced'' direction constructed by applying two linear transforms to the parametric components. These transforms give certain bases both in the subspace of gradients of ``active'' constraints and its complement in \(E^n\). The stepsize is chosen by the condition that the payoff function decreases, for which various penalty functions are taken. The parameters of the ``reduced'' direction are chosen to guarantee that the non-differentiable penalty payoff function decreases. In addition, a group of methods of linearization type were constructed. On the base of a modified linearization, the ``reduced'' direction was determined so that the objective function decreased along it. In this way methods of feasible directions were obtained of various orders. In the present article we suggest to use the differentiable penalty function \[ \Phi_\rho(x)= f_0(x)+ {1\over \rho} \sum_{j\in J} [\max\{0, f_j(x)\}]^2,\tag{3} \] as payoff function. We formulate the algorithm resulting from this penalty function and show its convergence.