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Polynomial Optimization

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Convex Analysis and Global Optimization

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 110))


Polynomial optimization is concerned with optimization problems described by multivariate polynomials on \(\mathbb{R}_{+}^{n}.\) In this chapter two approaches are presented for polynomial optimization. In the first approach a polynomial optimization problem is solved as a nonconvex optimization problem by a rectangular branch and bound algorithm in which bounding is performed by linear or convex relaxation. In the second approach, by viewing any multivariate polynomial on \(\mathbb{R}_{+}^{n}\) as a difference of two increasing functions, a polynomial optimization problem is treated as a monotonic optimization problem. In particular, the Successive Incumbent Transcending algorithm is developed which starts from a quickly found feasible solution then proceeds to gradually improving it to optimality.

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  • Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization. MPS-SIAM Series on Optimization. SIAM, Philadelphia (2001)

    Book  MATH  Google Scholar 

  • Duffin, R.J., Peterson, E.L., Zener, C.: Geometric Programming—Theory and Application. Wiley, New York (1967)

    MATH  Google Scholar 

  • Horst, R., Tuy, H.: Global Optimization (Deterministic Approaches), 3rd edn. Springer, Berlin/Heidelberg/New York (1996)

    Book  MATH  Google Scholar 

  • Lasserre, J.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11, 796–817 (2001)

    Google Scholar 

  • Lasserre, J.: Semidefinite programming vs. LP relaxations for polynomial programming. Math. Oper. Res. 27, 347–360 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42, 969–984 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Sherali, H.D., Adams, W.P.: A Reformulation-Lineralization Technique (RLT) for Solving Discrete and Continuous Nonconvex Programming Problems. Kluwer, Dordrecht (1999)

    Book  Google Scholar 

  • Shor, N.Z. : Quadratic Optimization Problems. Technicheskaya Kibernetica, 1, 128–139 (1987, Russian)

    Google Scholar 

  • Shor, N.Z., Stetsenko, S.I.: Quadratic Extremal Problems and Nondifferentiable Optimization. Naukova Dumka, Kiev (1989, Russian)

    Google Scholar 

  • Tuy, H.: Monotonic Optimization: Problems and Solution Approaches. SIAM J. Optim. 11, 464–494 (2000a)

    Google Scholar 

  • Tuy, H.: On dual bound methods for nonconvex global optimization. J. Glob. Optim. 37, 321–323 (2007b)

    Google Scholar 

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Tuy, H. (2016). Polynomial Optimization. In: Convex Analysis and Global Optimization. Springer Optimization and Its Applications, vol 110. Springer, Cham.

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