• 대한산업공학회지
• /
• 제27권3호
• /
• pp.274-280
• /
• 2001

In this research, we develop a specialized primal-dual interior point solver for the multicommodity flow problems (MCFP). The Castro's approach that exploits the problem structure is investigated and several aspects that must be considered in the implementation are addressed. First, we show how preprocessing techniques for linear programming(LP) are adjusted for MCFP. Secondly, we develop a procedure that extracts a network structure from the general LP formulated MCFP. Finally, we consider how the special structure of the mutual capacity constraints is exploited. Results of comupational comparison between our solver and a general interior point solver are also included.

• East Asian mathematical journal
• /
• 제29권3호
• /
• pp.279-292
• /
• 2013

It is well known that each kernel function defines a primal-dual interior-point method(IPM). Most of polynomial-time interior-point algorithms for linear optimization(LO) are based on the logarithmic kernel function([2, 11]). In this paper we define a new eligible kernel function and propose a new search direction and proximity function based on this function for LO problems. We show that the new algorithm has ${\mathcal{O}}((log\;p){\sqrt{n}}\;log\;n\;log\;{\frac{n}{\epsilon}})$ and ${\mathcal{O}}((q\;log\;p)^{\frac{3}{2}}{\sqrt{n}}\;log\;{\frac{n}{\epsilon}})$ iteration bound for large- and small-update methods, respectively. These are currently the best known complexity results.

• 한국경영과학회지
• /
• 제24권1호
• /
• pp.1-11
• /
• 1999

We classified preprocessing methods into (1) analytic methods, (2) methods for removing implied free variables, (3) methods using pivot or elementary row operations, (4) methods for removing linearly dependent rows and columns and (5) methods for dense columns. We noted some considerations to which should be paid attention when preprocessing methods are applied to interior point methods for linear programming. We proposed an efficient order of preprocessing methods and data structures. We also noted the recovery process for dual solutions. We implemented the proposed preprocessing methods. and tested it with 28 large scale problems of NETLIB. We compared the results of it with those of preprocessing routines of HOPDM, BPDPM and CPLEX.

• 대한수학회논문집
• /
• 제32권3호
• /
• pp.765-778
• /
• 2017

Finding an initial feasible solution on the central path is the main difficulty of feasible interior-point methods. Although, some algorithms have been suggested to remedy this difficulty, many practical implementations often do not use perfectly centered starting points. Therefore, it is worth to analyze the case that the starting point is not exactly on the central path. In this paper, we propose a weighted-path following interior-point algorithm for solving the Cartesian $P_{\ast}({\kappa})$-linear complementarity problems (LCPs) over symmetric cones. The convergence analysis of the algorithm is shown and it is proved that the algorithm terminates after at most $O\((1+4{\kappa}){\sqrt{r}}{\log}{\frac{x^0{\diamond}s^0}{\varepsilon}}\)$ iterations.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• 제12권4호
• /
• pp.227-238
• /
• 2008

In this paper we extend primal-dual interior point algorithm for linear optimization (LO) problems to $P_*({\kappa})$ linear complementarity problems(LCPs) ([1]). We define proximity functions and search directions based on kernel functions, ${\psi}(t)=\frac{t^{p+1}-1}{p+1}-{\log}\;t$, $p{\in}$[0, 1], which is a generalized form of the one in [16]. It is the first to use this class of kernel functions in the complexity analysis of interior point method(IPM) for $P_*({\kappa})$ LCPs. We show that if a strictly feasible starting point is available, then new large-update primal-dual interior point algorithms for $P_*({\kappa})$ LCPs have $O((1+2{\kappa})nlog{\frac{n}{\varepsilon}})$ complexity which is similar to the one in [16]. For small-update methods, we have $O((1+2{\kappa})\sqrt{n}{\log}{\frac{n}{\varepsilon}})$ which is the best known complexity so far.

• 한국실내디자인학회논문집
• /
• 제1호
• /
• pp.67-71
• /
• 1992

This study analyse the quantitative relationships between the compositive factors of the residential interior space in the view point of physical aspects. The proposed analytical methods are focused on finding the quantitative relationships between the human dimensions of the static and dynamic life style situations and the compositive factors of the architectural interior spaces. As a result, the major analystical methods are ; the analysis of Space-human Body System for finding the quantitative relationships between the human dimensions and the order of architectural structures ; the analysis of Space-Objects-Human Body System for finding the quantitiative relationships between the human dimensions and furniture dimensions of the interior space ; the analysis of Space-Objects - Life Activities of Human Body System for finding the quantitative relation ships between the human dimensions of daily or non daily life activities including static furniture systems.

• 한국경영과학회지
• /
• 제26권2호
• /
• pp.13-46
• /
• 2001

SDP (Semidefinite Programming), as a sort of “cone-LP”, optimizes a linear function over the intersection of an affine space and a cone that has the origin as its apex. SDP, however, has been developed in the process of searching for better solution methods for NP-hard combinatorial optimization problems. We surveyed the basic theories necessary to understand SDP researches. First, We examined SDP duality, comparing it to LP duality, which is essential for the interior point method, Second, we showed that SDP can be optimized from an interior solution in polynomial time with a desired error limit. finally, we summarized several research papers that showed SDP can improve solution methods for some combinatorial optimization problems, and explained why SDP has become one of the most important research topics in optimization. We tried to integrate SDP theories. relatively diverse and complicated. to survey research papers with our own perspective, and thus to help researcher to pursue their SDP researches in depth.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• 제14권2호
• /
• pp.93-111
• /
• 2010

The purpose of this paper is to obtain new complexity results for a second-order cone optimization (SOCO) problem. We define a proximity function for the SOCO by a kernel function. Furthermore we formulate an algorithm for a large-update primal-dual interior-point method (IPM) for the SOCO by using the proximity function and give its complexity analysis, and then we show that the new worst-case iteration bound for the IPM is $O(q\sqrt{N}(logN)^{\frac{q+1}{q}}log{\frac{N}{\epsilon})$, where $q{\geqq}1$.

• East Asian mathematical journal
• /
• 제26권1호
• /
• pp.9-23
• /
• 2010

In this paper we generalize large-update primal-dual interior point methods for linear optimization problems in [2] to the $P_*(\kappa)$ linear complementarity problems based on a new kernel function which includes the kernel function in [2] as a special case. The kernel function is neither self-regular nor eligible. Furthermore, we improve the complexity result in [2] from $O(\sqrt[]{n}(\log\;n)^2\;\log\;\frac{n{\mu}o}{\epsilon})$ to $O\sqrt[]{n}(\log\;n)\log(\log\;n)\log\;\frac{m{\mu}o}{\epsilon}$.

• Journal of applied mathematics & informatics
• /
• 제29권5_6호
• /
• pp.1245-1256
• /
• 2011

In this paper, we proposed a regularization interior point method for semidefinite programming with free variables which can be taken as an extension of the algorithm for standard semidefinite programming. Since an inexact search direction at each iteration is used, the computation of the designed algorithm is much less compared with the existing solution methods. The convergence analysis of the method is established under weak conditions.