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An algorithm for linear programming which requires O(((m+n)n 2 +(m+n) 1·5 n)L) arithmetic operations. (English) Zbl 0708.90047
The author gives an algorithm for solving a linear programming problem max{c T x; Axb} where A is an m×n matrix. The algorithm employs the idea due to J. Renegar [Math. Program., Ser. A 40, No.1, 59-93 (1988; Zbl 0654.90050)] and proceeds by constructing a sequence of smaller and smaller polytopes which shrink towards the optimal vertex. At each iteration the algorithm moves from the center of the current polytope to the center of the next one by minimizing a linear function over an ellipsoid. The total number of arithmetic operations required is O(((m+n)n 2 +(m+n) 1·5 n)L), where L is bounded by the number of bits in the input; thus the algorithm is faster than that of N. Karmarkar [Combinatorica 4, 373-395 (1984; Zbl 0557.90065)] by a factor of m.
Reviewer: J.Rohn
MSC:
90C05Linear programming
90C60Abstract computational complexity for mathematical programming problems
52B12Special polytopes (linear programming, centrally symmetric, etc.)
References:
[1]D.A. Bayer and J.C. Lagarias, ”The non-linear geometry of linear programming I. Affine and projective scaling trajectories,”Transactions of the American Mathematical Society (1989), to appear.
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[4]C. Gonzaga, ”An algorithm for solving linear programming problems in O(n 3 L) operations,” Memorandum UCB/ERL M87/10, Electronics Research Laboratory, University of Berkeley (Berkeley, CA, 1987).
[5]N. Karmarkar, ”A new polynomial time algorithm for linear programming,”Combinatorica 4 (1984) 373–395. · Zbl 0557.90065 · doi:10.1007/BF02579150
[6]L.G. Khachian, ”Polynomial algorithms in linear programming,”Žurnal Vyčislitel’noî Matematiki i Matematičeskoî Fiziki 20 (1980) 53–72.
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[8]J. Renegar, ”A polynomial-time algorithm, based on Newton’s method, for linear programming,”Mathematical Programming, 40 (1988) 59–93. · Zbl 0654.90050 · doi:10.1007/BF01580724
[9]Gy. Sonnevand, ”An analytical center for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming,” preprint, Department of Numerical Analysis, Institute of Mathematics, Eötvös University (Budapest, 1989) 6–8.
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[11]P.M. Vaidya, ”An algorithm for linear programming which requires O(((m+n)n 2+(m+n)1.5 n)L) arithmetic operations,”Proceedings 19th Annual ACM Symposium Theory of Computing (1987) 29–38.
[12]J.H. Wilkinson,The Algebraic Eigenvalue Problem (Oxford University Press (Clarendon), London and New York, 1965).
[13]G. Zoutendijk,Mathematical Programming Methods (North-Holland, New York, 1976).