Disciplined quasiconvex programming

We present a composition rule involving quasiconvex functions that generalizes the classical composition rule for convex functions. This rule complements well-known rules for the curvature of quasiconvex functions under increasing functions and pointwise maximums. We refer to the class of optimization problems generated by these rules, along with a base set of quasiconvex and quasiconcave functions, as disciplined quasiconvex programs . Disciplined quasiconvex programming generalizes disciplined convex programming, the class of optimization problems targeted by most modern domain-specific languages for convex optimization. We describe an implementation of disciplined quasiconvex programming that makes it possible to specify and solve quasiconvex programs in CVXPY 1.0.

[1]  François Laviolette,et al.  Domain-Adversarial Training of Neural Networks , 2015, J. Mach. Learn. Res..

[2]  Shai Shalev-Shwartz,et al.  Beyond Convexity: Stochastic Quasi-Convex Optimization , 2015, NIPS.

[3]  Daniel Liberzon,et al.  Quantized control via locational optimization , 2002, IEEE Transactions on Automatic Control.

[4]  Harvey J. Greenberg,et al.  A Review of Quasi-Convex Functions , 1971, Oper. Res..

[5]  K. Gu.  Designing stabilizing control of uncertain systems by quasiconvex optimization , 1994, IEEE Trans. Autom. Control..

[6]  H. Nikaidô.  On von Neumann’s minimax theorem , 1954 .

[7]  J. Lofberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[8]  Takeo Kanade,et al.  Uncertainty Models in Quasiconvex Optimization for Geometric Reconstruction , 2006, 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'06).

[9]  K. Arrow,et al.  EXISTENCE OF AN EQUILIBRIUM FOR A COMPETITIVE ECONOMY , 1954 .

[10]  Elena Molho,et al.  The origins of quasi-concavity: a development between mathematics and economics , 2004 .

[11]  David Eppstein,et al.  Quasiconvex Programming , 2004, ArXiv.

[12]  Robert Kantrowitz,et al.  Optimization for products of concave functions , 2005 .

[13]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[14]  W. Fenchel.  Convex cones, sets, and functions , 1953 .

[15]  David G. Luenberger,et al.  Quasi-Convex Programming , 1968 .

[16]  Alexandre Megretski,et al.  A Quasi-Convex Optimization Approach to Parameterized Model Order Reduction , 2005, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[17]  Krzysztof C. Kiwiel,et al.  Convergence and efficiency of subgradient methods for quasiconvex minimization , 2001, Math. Program..

[18]  Takeo Kanade,et al.  Quasiconvex Optimization for Robust Geometric Reconstruction , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[19]  Stephen P. Boyd,et al.  Disciplined Convex Programming , 2006 .

[20]  Stephen Boyd,et al.  A Rewriting System for Convex Optimization Problems , 2017, ArXiv.

[21]  Stephen P. Boyd,et al.  Convex Optimization in Julia , 2014, 2014 First Workshop for High Performance Technical Computing in Dynamic Languages.

[22]  Igor V. Konnov,et al.  On Convergence Properties of a Subgradient Method , 2003, Optim. Methods Softw..

[23]  Alfred V. Aho,et al.  Compilers: Principles, Techniques, and Tools , 1986, Addison-Wesley series in computer science / World student series edition.

[24]  S. Schaible.  Fractional programming: Applications and algorithms , 1981 .

[25]  S. Zionts,et al.  Programming with linear fractional functionals , 1968 .

[26]  Alfred V. Aho,et al.  Compilers: Principles, Techniques, and Tools (2nd Edition) , 2006 .

[27]  Stephen P. Boyd,et al.  CVXR: An R Package for Disciplined Convex Optimization , 2017, Journal of Statistical Software.

[28]  Peter Seiler,et al.  Quasiconvex sum-of-squares programming , 2010, 49th IEEE Conference on Decision and Control (CDC).

[29]  C. R. Bector.  Programming Problems with Convex Fractional Functions , 1968, Oper. Res..

[30]  Stephen P. Boyd,et al.  CVXPY: A Python-Embedded Modeling Language for Convex Optimization , 2016, J. Mach. Learn. Res..