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Absolute Value Equation Solution via Concave Minimization

Absolute Value Equation Solution via Concave Minimization. Olvi Mangasarian University of Wisconsin - Madison University of California at San Diego. What is the Absolute Value Equation (AVE)?. Simplest Solvable AVE in 1-Dimension. Why the Absolute Value Equation ?. Because the AVE subsumes:

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Absolute Value Equation Solution via Concave Minimization

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  1. Absolute Value Equation SolutionviaConcave Minimization Olvi Mangasarian University of Wisconsin - Madison University of California at San Diego

  2. What is the Absolute Value Equation (AVE)?

  3. Simplest Solvable AVE in 1-Dimension

  4. Why the Absolute Value Equation ? • Because the AVE subsumes: • Linear Programming • Quadratic Progamming • Bimatrix games • The Linear complementarity problem: which is NP-complete in its general form

  5. Some Existence Results for the AVEOLM & R. R. Meyer 2005 • The AVE is uniquely solvable for any b2 Rn if the singular values of A exceed 1. • The AVE is uniquely solvable for any b2 Rn ifk A-1k <1. • Let A ¸ 0, k A k <1 and b·0, then a nonnegative solution of the AVE exists. • The AVE has no solution ifk A k <1 and 0 b¸ 0.

  6. Why is the AVE NP-Hard? • Since each of n components of x can be positive or nonpositive in Ax-|x|=b • One needs to look at 2n possibilities • Thus for n=1000 we have 21000 such possibilities (21000 > 109) • We will attempt to solve 100 such 1000-dimensional randomly generated solvable AVEs • We will minimize a piecewise-linear concave function on a polyhedral set using linear programming to solve successive linearizations of the concave function

  7. Brief Review of NP-Hard & NP-Complete Problems • Problems I & II are polynomially equivalent if a polynomial time algorithm solves I implies the same for II & conversely • NP-complete problem is one that is polynomially equivalent to any one of the standard intractable problems such as the traveling salesman problem, partition problem, norm maximization problem or the linear complementarity problem • Nondeterministic algorithm is an algorithm with more than one allowed choice at each step and which always makes the right or best choice at each step and solves the problem in a finite number of steps • NP is class of problems solvable by a nondeterministic algorithm in polynomial time including NP-complete problems • NP-hard problem is one such that an NP-complete problem reduces to it in polynomial time

  8. R. T.Rockafellar: “Convex Analysis” 1970 Corollaries 32.3.3 & 32.3.4

  9. AVE as Piecewise Linear Concave Minimization

  10. Proof of Piecewise Linear Concave Minimization

  11. Existence of a Vertex Solution

  12. Existence of a Vertex Solution(Proof)

  13. Existence of a Vertex Solution (Proof Continued)

  14. The Successive Linearization Algorithm (SLA)

  15. SLA Example

  16. SLA Example

  17. SLA Example

  18. SLA Finite Termination Theorem

  19. SLA MATLAB Code

  20. Computational ResultsGlossary of Terms nztot: total number of violated equations in each group of 10 problems nnzx: maximum number of violated equations per individual problem in the group of 10 problems k: total number of iterations (LPs) utilized for each Group of 10 problems minutes: total time for solving group of 10 problems minutes per LP: average time for each LP

  21. Computational Results100 Instances of 1000 Dimensional Problems

  22. Summary of Computational Results • Generated 100 consecutive random instances of 1000 dimensional solvable AVEs • Out of 100 instances, 95 instances were solved to an accuracy of 1e-6 • For each of the 5 unsolved instances, only one equation out of a thousand equations were violated per instance • The average number of LPs per instance varied from 2.6 to 9.1 with the overall average for the 100 problems being 4.81 LPS per instance • The overall average time for solving each instance of 1000 fully dense equations was 4.13 minutes on a 3.00GHz Pentium 4 processor running i386_tao10 Linux • The average time to solve each LP was 0.86 minutes

  23. Conclusion & Outlook • AVE: Ax-|x|=b is a fascinatingly simple problem that is NP-hard • We have proposed an LP-based SLA method of solution that worked on 95 out of 100 consecutively generated 1000-dimensional solvable random problems • Possible future work: • Investigate other algorithms • Improve proposed algorithm • Apply AVE formulation to known related problems and to other problems

  24. Related Paper • ``Solution of general linear complementarity problems via • nondifferentiable concave minimization". Mathematical • Programming Technical Report 96-10, November 1996. • Acta Mathematica Vietnamica, 22(1), 1997, 199-205. Talk & Paper on Web • http://www.cs.wisc.edu/~olvi • Optimization Letters 1, 2006. To appear.

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