A generalized interior-point barrier function approach for smooth convex programming with linear constraints

Interior point methods were originally designed to obtain the polynomial-time complexity for linear programming. Extension to convex quadratic programming is straightforward because, when we take the first derivative of the quadratic function for use in KKT conditions, it becomes linear. However, further extension to convex programming remains a difficult task due to the nonlinearity. Here in this paper, we demonstrate a systematic way of interior point approach for solving this problem. General type of barrier functions are studied and conditions which ensure convergence and polynomiality are exploited. As an interesting special case, we look into the logarithmic barrier function, and propose a standard Lipschitz notion for local analysis. A phase I procedure is also included to make this work complete.