Relaxed cutting plane method with convexification for solving nonlinear semi-infinite programming problems

In this paper, we present an algorithm to solve nonlinear semi-infinite programming (NSIP) problems. To deal with the nonlinear constraint, Floudas and Stein (SIAM J. Optim. 18:1187–1208, 2007) suggest an adaptive convexification relaxation to approximate the nonlinear constraint function. The αBB method, used widely in global optimization, is applied to construct the convexification relaxation. We then combine the idea of the cutting plane method with the convexification relaxation to propose a new algorithm to solve NSIP problems. With some given tolerances, our algorithm terminates in a finite number of iterations and obtains an approximate stationary point of the NSIP problems. In addition, some NSIP application examples are implemented by the method proposed in this paper, such as the proportional-integral-derivative controller design problem and the nonlinear finite impulse response filter design problem. Based on our numerical experience, we demonstrate that our algorithm enhances the computational speed for solving NSIP problems.

[1]  R. Reemtsen,et al.  The Direct Solution of Nonconvex Nonlinear FIR Filter Design Problems by a SIP Method , 2000 .

[2]  Jane J. Ye,et al.  Solving semi-infinite programs by smoothing projected gradient method , 2014, Comput. Optim. Appl..

[3]  Kok Lay Teo,et al.  A Truncated Projected Newton-Type Algorithm for Large-Scale Semi-infinite Programming , 2006, SIAM J. Optim..

[4]  Paul I. Barton,et al.  Interval Methods for Semi-Infinite Programs , 2005, Comput. Optim. Appl..

[5]  Dong-Hui Li,et al.  An iterative method for solving KKT system of the semi-infinite programming , 2005, Optim. Methods Softw..

[6]  Kenneth O. Kortanek,et al.  Semi-Infinite Programming: Theory, Methods, and Applications , 1993, SIAM Rev..

[7]  Oliver Stein,et al.  The Adaptive Convexification Algorithm: A Feasible Point Method for Semi-Infinite Programming , 2007, SIAM J. Optim..

[8]  Marco A. López,et al.  A New Exchange Method for Convex Semi-Infinite Programming , 2010, SIAM J. Optim..

[9]  Kenneth O. Kortanek,et al.  A Central Cutting Plane Algorithm for Convex Semi-Infinite Programming Problems , 1993, SIAM J. Optim..

[10]  Georg Still,et al.  Discretization in semi-infinite programming: the rate of convergence , 2001, Math. Program..

[11]  C. Adjiman,et al.  A global optimization method, αBB, for general twice-differentiable constrained NLPs—II. Implementation and computational results , 1998 .

[12]  Aurelio Piazzi,et al.  A hybrid algorithm for infinitely constrained optimization , 2001, Int. J. Syst. Sci..

[13]  Kok Lay Teo,et al.  Computational Discretization Algorithms for Functional Inequality Constrained Optimization , 2000, Ann. Oper. Res..

[14]  Masao Fukushima,et al.  A globally convergent SQP method for semi-infinite nonlinear optimization , 1988 .

[15]  Bruno Betrò,et al.  An accelerated central cutting plane algorithm for linear semi-infinite programming , 2004, Math. Program..

[16]  Marco A. López,et al.  Semi-infinite programming , 2007, Eur. J. Oper. Res..

[17]  Alexander Potchinkov,et al.  THE SIMULTANEOUS APPROXIMATION OF MAGNITUDE AND PHASE BY FIR DIGITAL FILTERS. II: METHODS AND EXAMPLES , 1997 .

[18]  C. C. Gonzaga,et al.  An improved algorithm for optimization problems with functional inequality constraints , 1980 .

[19]  Ekkehard W. Sachs,et al.  Local Convergence of SQP Methods in Semi-Infinite Programming , 1995, SIAM J. Optim..

[20]  Paul I. Barton,et al.  Global solution of semi-infinite programs , 2004 .

[21]  Kok Lay Teo,et al.  A computational algorithm for functional inequality constrained optimization problems , 1990, Autom..

[22]  A. Neumaier,et al.  A global optimization method, αBB, for general twice-differentiable constrained NLPs — I. Theoretical advances , 1998 .

[23]  Christodoulos A. Floudas,et al.  Deterministic Global Optimization: Theory, Methods and (NONCONVEX OPTIMIZATION AND ITS APPLICATIONS Volume 37) (Nonconvex Optimization and Its Applications) , 2005 .

[24]  Christodoulos A. Floudas,et al.  Deterministic global optimization - theory, methods and applications , 2010, Nonconvex optimization and its applications.

[25]  Alexander Potchinkov,et al.  The Simultaneous Approximation of Magnitude and Phase by FIR Digital Filters. I: A New Approach , 1997, Int. J. Circuit Theory Appl..

[26]  C. J. Price,et al.  Numerical experiments in semi-infinite programming , 1996, Comput. Optim. Appl..

[27]  Alexander Potchinkov,et al.  FIR filter design in the complex domain by a semi-infinite programming technique. I: The method , 1994 .

[28]  Elijah Polak,et al.  Optimization: Algorithms and Consistent Approximations , 1997 .

[29]  A. Shapiro.  Semi-infinite programming, duality, discretization and optimality conditions , 2009 .

[30]  Chen Ling,et al.  A smoothing projected Newton-type algorithm for semi-infinite programming , 2009, Comput. Optim. Appl..

[31]  Rembert Reemtsen,et al.  Numerical Methods for Semi-Infinite Programming: A Survey , 1998 .