×

zbMATH — the first resource for mathematics

Strict quasi-concavity and the differential barrier property of gauges in linear programming. (English) Zbl 1327.90100
Summary: Concave gauge functions were introduced to give an analytical representation of cones. In particular, they give a simple and a practical representation of the positive orthant. There is a wide choice of concave gauge functions with interesting properties, representing the same cone. Besides the fact that a concave gauge cannot be identically zero on a cone(\(\neq\{0\}\)), it may be continuous, differentiable and even \(\mathcal{C}^\infty\) on its interior. The purpose of the present paper is to present another approach to penalizing the positivity constraints of a linear programme using an arbitrary strictly quasi-concave gauge representation. Throughout the paper, we generalize the concept of the central path and the analytic centre in terms of these gauges, introduce the differential barrier concept and establish its relationship with strict quasi-concavity.

MSC:
90C05 Linear programming
90C51 Interior-point methods
49M30 Other numerical methods in calculus of variations (MSC2010)
49N15 Duality theory (optimization)
Software:
LIPSOL; PCx
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1007/BF02579150 · Zbl 0557.90065 · doi:10.1007/BF02579150
[2] Zhang Y. 1995. User’s Guide to LIPSOL. Technical Report TR95-19. Baltimore, MD: Department of Mathematics and statistics, University of Maryland Baltimore County.
[3] Zhang Y, Solving large-scale linear programs by interior-point methods under the MATLAB environment, Technical Report TR96-01 (1996)
[4] Czyzyk J, Mehrotra S, Wagner M, Wright SJ. PCx User Guide (Version 1.1), Technical Report OTC 96/01; 1997.
[5] Mehrotra S. Higher order method and their performance. Technical Report 90-16R1. Evanson, IL (USA): Dept. of Industrial Engineering and Management Science, Northwestern University; 1990. Revised July 1991.
[6] DOI: 10.1137/0802028 · Zbl 0773.90047 · doi:10.1137/0802028
[7] DOI: 10.1007/978-1-4613-9617-8_2 · doi:10.1007/978-1-4613-9617-8_2
[8] Frisch KR, The logarithmic potential method of convex programming. Technical report (1955)
[9] DOI: 10.1007/978-1-4757-3216-0_18 · doi:10.1007/978-1-4757-3216-0_18
[10] DOI: 10.1007/978-1-4613-3449-1_6 · doi:10.1007/978-1-4613-3449-1_6
[11] DOI: 10.1137/1034048 · Zbl 0763.90063 · doi:10.1137/1034048
[12] Roos C, Interior point methods for mathematical programming (1997)
[13] DOI: 10.1007/978-0-387-74388-2 · doi:10.1007/978-0-387-74388-2
[14] Barbara A, Z. Oper. Res 40 pp 43– (1994)
[15] Penot JP, J. Convex Anal 7 pp 95– (2000)
[16] DOI: 10.1007/978-1-4757-3200-9 · doi:10.1007/978-1-4757-3200-9
[17] Martinez-Legaz JE, Acta Math. Vietnamica 26 pp 313– (2001)
[18] Zaffaroni A, J. Convex Anal 15 pp 325– (2008) · Zbl 1148.52002
[19] DOI: 10.1515/9781400873173 · Zbl 0932.90001 · doi:10.1515/9781400873173
[20] DOI: 10.1007/978-3-642-95342-2_1 · doi:10.1007/978-3-642-95342-2_1
[21] Tind J. Blocking and antiblocking sets. Vol. 6, Mathematical programming. Amsterdam: North Holland Publishing Company; 1974. p. 157–166. · Zbl 0288.90074
[22] DOI: 10.1007/978-3-642-02431-3 · Zbl 0888.49001 · doi:10.1007/978-3-642-02431-3
[23] Fiacco AV, Nonlinear programming: sequential unconstrained minimization techniques (1968) · Zbl 0193.18805
[24] DOI: 10.1007/s10898-012-9918-z · Zbl 1312.90079 · doi:10.1007/s10898-012-9918-z
[25] DOI: 10.1090/S0002-9904-1943-07818-4 · Zbl 0063.00985 · doi:10.1090/S0002-9904-1943-07818-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.