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 \(h\in S\), \(+\epsilon^ q\leq 0\) for \(j\in J_ 1(x_ k,\epsilon^ p)\), \(\leq 0\) 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.