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Feasible Direction Interior-Point Technique for Nonlinear Optimization

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  • J. Herskovits

    (Federal University of Rio de Janeiro)

Abstract

We propose a feasible direction approach for the minimization by interior-point algorithms of a smooth function under smooth equality and inequality constraints. It consists of the iterative solution in the primal and dual variables of the Karush–Kuhn–Tucker first-order optimality conditions. At each iteration, a descent direction is defined by solving a linear system. In a second stage, the linear system is perturbed so as to deflect the descent direction and obtain a feasible descent direction. A line search is then performed to get a new interior point and ensure global convergence. Based on this approach, first-order, Newton, and quasi-Newton algorithms can be obtained. To introduce the method, we consider first the inequality constrained problem and present a globally convergent basic algorithm. Particular first-order and quasi-Newton versions of this algorithm are also stated. Then, equality constraints are included. This method, which is simple to code, does not require the solution of quadratic programs and it is neither a penalty method nor a barrier method. Several practical applications and numerical results show that our method is strong and efficient.

Suggested Citation

  • J. Herskovits, 1998. "Feasible Direction Interior-Point Technique for Nonlinear Optimization," Journal of Optimization Theory and Applications, Springer, vol. 99(1), pages 121-146, October.
    • Handle: RePEc:spr:joptap:v:99:y:1998:i:1:d:10.1023_a:1021752227797
    DOI: 10.1023/A:1021752227797
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    References listed on IDEAS

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    1. Renato D. C. Monteiro & Ilan Adler & Mauricio G. C. Resende, 1990. "A Polynomial-Time Primal-Dual Affine Scaling Algorithm for Linear and Convex Quadratic Programming and Its Power Series Extension," Mathematics of Operations Research, INFORMS, vol. 15(2), pages 191-214, May.
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    Cited by:

    1. A. F. Izmailov & M. V. Solodov, 2015. "Newton-Type Methods: A Broader View," Journal of Optimization Theory and Applications, Springer, vol. 164(2), pages 577-620, February.
    2. Carolina Effio Saldivar & José Herskovits & Juan Pablo Luna & Claudia Sagastizábal, 2019. "Multidimensional Calibration Of Crude Oil And Refined Products Via Semidefinite Programming Techniques," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(01), pages 1-31, February.
    3. Napsu Karmitsa & Mario Tanaka Filho & José Herskovits, 2011. "Globally Convergent Cutting Plane Method for Nonconvex Nonsmooth Minimization," Journal of Optimization Theory and Applications, Springer, vol. 148(3), pages 528-549, March.
    4. Angel E. R. Gutierrez & Sandro R. Mazorche & José Herskovits & Grigori Chapiro, 2017. "An Interior Point Algorithm for Mixed Complementarity Nonlinear Problems," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 432-449, November.
    5. Grigori Chapiro & Angel E. R. Gutierrez & José Herskovits & Sandro R. Mazorche & Weslley S. Pereira, 2016. "Numerical Solution of a Class of Moving Boundary Problems with a Nonlinear Complementarity Approach," Journal of Optimization Theory and Applications, Springer, vol. 168(2), pages 534-550, February.
    6. Stefan C. Endres & Carl Sandrock & Walter W. Focke, 2018. "A simplicial homology algorithm for Lipschitz optimisation," Journal of Global Optimization, Springer, vol. 72(2), pages 181-217, October.
    7. Alfredo Canelas & Miguel Carrasco & Julio López, 2017. "Application of the sequential parametric convex approximation method to the design of robust trusses," Journal of Global Optimization, Springer, vol. 68(1), pages 169-187, May.

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