Rate of convergence of the method of feasible directions, not necessarily using the direction of steepest descent (Q1119473)

The author proposes an extension of the feasible direction method for the convex programming problem: \(\min \{f^ 0(x)\), \(x\in D\}\), where \(D=\{x\in {\mathbb{R}}^ n\), \(f_ j(x)\leq 0\), \(j\in J\}\), \(J=\{1,...,m\}\). The extension is as follows. (0) Choose \(\epsilon_ 0>0\), \(\beta\in (0,1)\), \(p>0\), \(q>0.\) (1) Find \(x_ 0\in D\), set \(k=0\), \(\epsilon =\epsilon_ 0.\) (2) Choose \(h(x_ k,\epsilon)\) such that \(h\in S\), \(<f_ j'(x_ k),h>+\epsilon^ q\leq 0\) for \(j\in J_ 1(x_ k,\epsilon^ p)\), \(<f_ j'(x_ k),h>\leq 0\) for \(j\in J_ 2(x_ k,\epsilon^ p)\), where \(J_ 2=\{j\in J\), \(f_ j'(x)=const\}\), \(J_ 1=J\setminus J_ 2\), \(J(x,\epsilon)=I(x,\epsilon)\cup \{0\}\), \(I(x,\epsilon)=\{j\in J\), \(f_ j(x)\geq -\epsilon \}\), S is a compact set in \({\mathbb{R}}^ n\) containing the origin in its interior. (3) If there is no such h set \(\epsilon:=\beta \epsilon\) and go to (2). (4) Find \(x_{k+1}\) by a line search along h in D. The author obtains an estimate of the rate of convergence using assumptions such as boundedness of the level set of \(f^ 0\), Lipschitz continuity and convexity of the functions, Slater condition for the nonlinear constraints, uniqueness of the solution, etc.