• Journal of Advanced Marine Engineering and Technology
• /
• v.28 no.5
• /
• pp.801-810
• /
• 2004

The Primal-Dual Interior-Point (PDIP) method is currently one of the fastest emerging topics in optimization. This method has become an effective solution algorithm for large scale nonlinear optimization problems. such as the electric Optimal Power Flow (OPF) and natural gas and electricity OPF. This study describes major theoretical developments of the PDIP method as well as practical issues related to implementation of the method. A simple quadratic problem with linear equality and inequality constraints

• Proceedings of the Korean Operations and Management Science Society Conference
• /
• 1996.04a
• /
• pp.24-29
• /
• 1996

If the LP problem doesn't have the optimal soultion uniquely, the solution fo the primal-dual barrier method converges to the interior point of the optimal face. Therefore, when the optimal vertex solution or the optimal basis is required, we have to perform the additional procedure to recover the optimal basis from the final solution of the interior point method. In this paper the exisiting methods for recovering the optimal basis or identifying the optimal solutions are analyzed and the new methods are suggested. This paper treats the two optimal basis recovery methods. One uses the purification scheme and the simplex method, the other uses the optimal face solutions. In the method using the purification procedure and the simplex method, the basic feasible solution is obtained from the given interior solution and then simplex method is performed for recovering the optimal basis. In the method using the optimal face solutions, the optimal basis in the primal-dual barrier method is constructed by intergrating the optimal solution identification technique and the optimal basis extracting method from the primal-optimal soltion and the dual-optimal solution.

• Korean Management Science Review
• /
• v.11 no.3
• /
• pp.1-11
• /
• 1994

Recent advances in linear programming solution methodology have focused on interior point methods. This powerful new class of methods achieves significant reductions in computer time for large linear programs and solves problems significantly larger than previously possible. These methods can be examined from points of Fiacco and McCormick's barrier method, Lagrangian duality, Newton's method, and others. This study presents a primal-dual log barrier algorithm of interior point methods for linear programming. The primal-dual log barrier method is currently the most efficient and successful variant of interior point methods. This paper also addresses a Cholesky factorization method of symmetric positive definite matrices arising in interior point methods. A special structure of the matrices, called supernode, is exploited to use computational techniques such as direct addressing and loop-unrolling. Two dense matrix handling techniques are also presented to handle dense columns of the original matrix A. The two techniques may minimize storage requirement for factor matrix L and a smaller number of arithmetic operations in the matrix L computation.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.13 no.1
• /
• pp.41-53
• /
• 2009

A primal-dual interior point method(IPM) not only is the most efficient method for a computational point of view but also has polynomial complexity. Most of polynomialtime interior point methods(IPMs) are based on the logarithmic barrier functions. Peng et al.([14, 15]) and Roos et al.([3]-[9]) proposed new variants of IPMs based on kernel functions which are called self-regular and eligible functions, respectively. In this paper we define a new kernel function and propose a new IPM based on this kernel function which has $O(n^{\frac{2}{3}}log\frac{n}{\epsilon})$ and $O(\sqrt{n}log\frac{n}{\epsilon})$ iteration bounds for large-update and small-update methods, respectively.

• Journal of applied mathematics & informatics
• /
• v.27 no.5_6
• /
• pp.981-1000
• /
• 2009

This paper considers asset liability management problems when their deterministic equivalent formulations are general nonlinear optimization problems. The presented approach uses a nonmonotone trust region strategy for solving a sequence of unconstrained subproblems parameterized by a scalar parameter. The objective function of each unconstrained subproblem is an augmented penalty-barrier function that involves both primal and dual variables. Each subproblem is solved approximately. The algorithm does not restrict a monotonic decrease of the objective function value at each iteration. If a trial step is not accepted, the algorithm performs a non monotone line search to find a new acceptable point instead of resolving the subproblem. We prove that the algorithm globally converges to a point satisfying the second-order necessary optimality conditions.

• Proceedings of the Korean Operations and Management Science Society Conference
• /
• 1996.04a
• /
• pp.530-533
• /
• 1996

This paper deals with finding an initial solution and modifying search direction by the centrering force in the predictor-corrector method which is a variant of the primal-dual barrier method. These methods were tested with NETLIB problems. Initial solutions which are located close to the center of the feasible set lower the number of iterations, as they enlarge the step length. Three heuristic methods to find such initial solution are suggested. The new methods reduce the average number of iterations by 52% to at most, compared with the old method assigning 1 to initial valurs. Solutions can move closer to the central path fast by enlarging the centering force in early steps. It enlarge the step length, so reduces the number of iterations. The more effective this method is the closer the initial solution is to the boundary of the feasible set.

• Korean Management Science Review
• /
• v.21 no.1
• /
• pp.77-86
• /
• 2004

In this paper, we present a methodology for solving the multicommodity network flow problems using interior point methods. In our method, the minimum cost network flow problem extracted from the given multicommodity network flow problem is solved by primal-dual barrier method in which normal equations are solved partially using preconditioned conjugate gradient method. Based on the solution of the minimum cost network flow problem, a warm-start point is obtained from which Castro's specialized interior point method for multicommodity network flow problem starts. In the computational experiments, the effectiveness of our methodology is shown.

• Bulletin of the Korean Mathematical Society
• /
• v.53 no.6
• /
• pp.1831-1844
• /
• 2016

In this paper, we improve the full-Newton step infeasible interior-point algorithm proposed by Mansouri et al. [6]. The algorithm takes only one full-Newton step in a major iteration. To perform this step, the algorithm adopts the largest logical value for the barrier update parameter ${\theta}$. This value is adapted with the value of proximity function ${\delta}$ related to (x, y, s) in current iteration of the algorithm. We derive a suitable interval to change the parameter ${\theta}$ from iteration to iteration. This leads to more flexibilities in the algorithm, compared to the situation that ${\theta}$ takes a default fixed value.