Sequence of polyhedral relaxations for nonlinear univariate functions

@article{Sundar2020SequenceOP,
  title={Sequence of polyhedral relaxations for nonlinear univariate functions},
  author={K. Sundar and Sujeevraja Sanjeevi and Harsha Nagarajan},
  journal={arXiv: Optimization and Control},
  year={2020}
}
The letter develops a sequence of Mixed Integer Linear Programming (MILP) and Linear Programming (LP) relaxations that converge to the graph of a nonlinear, univariate, bounded, and differentiable function $f(x)$ and its convex hull, respectively. Theoretical convergence of the sequence of relaxations to the graph of the function and its convex hull is established. These relaxations can be used in MILP-based global optimization algorithms for nonlinear non-convex optimization problems. 

Figures and Tables from this paper

References

SHOWING 1-10 OF 46 REFERENCES
Convex Envelopes of Monomials of Odd Degree
TLDR
It is proved that this envelope for monomial terms of odd degree, x2k+1, is the tightest possible and derive a linear relaxation from the proposed envelope, and compare both the nonlinear and linear formulations with relaxations obtained using other approaches. Expand
AN ALGORITHMIC FRAMEWORK FOR MINLP WITH SEPARABLE NON-CONVEXITY
We present an algorithm for Mixed-Integer Nonlinear Programming (MINLP) problems in which the non-convexity in the objective and constraint functions is manifested as the sum of non-convex univariateExpand
Piecewise Polyhedral Formulations for a Multilinear Term
TLDR
This paper presents a mixed-integer linear programming (MILP) formulation of a piecewise, polyhedral relaxation (PPR) of a multilinear term using its convex hull representation and presents computational results showing the effectiveness of proposed formulations on instances of standard benchmarks of nonlinear programs (NLPs) with multILinear terms. Expand
Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming
TLDR
The GAMS/BARON Model for Pooling Problems is presented as a model for solving the refrigerant design problem and a number of examples show how this model can be applied to distributed systems. Expand
Global optimization of MIQCPs with dynamic piecewise relaxations
TLDR
A new deterministic global optimization algorithm for solving mixed-integer bilinear programs that alternates between piecewise McCormick and normalized multiparametric disaggregation and obtains the same or better optimality gaps than two commercial global optimization solvers. Expand
Error analysis for convex separable programs: Bounds on optimal and dual optimal solutions
Abstract Computable lower and upper bounds on the optimal and dual optimal solutions of a nonlinear, convex separable program are obtained from its piecewise linear approximation. They provideExpand
Computing tight bounds via piecewise linear functions through the example of circle cutting problems
  • S. Rebennack
  • Mathematics, Computer Science
  • Math. Methods Oper. Res.
  • 2016
TLDR
A new global optimization algorithm is introduced, based on piecewise linear function approximations, which converges in finitely many iterations to a globally optimal solution. Expand
Tight Piecewise Convex Relaxations for Global Optimization of Optimal Power Flow
TLDR
This work develops tight piecewise convex relaxations with convex-hull representations, an adaptive, multivariate partitioning algorithm with bound tightening that progressively improves these relaxations and converges to the globally optimal solution. Expand
Univariate parameterization for global optimization of mixed-integer polynomial problems
TLDR
The new underestimation approach can be made as tight as desired and is shown capable of providing considerably better lower bounds than a widely used global optimization solver for a specific class of design problems involving bilinear terms. Expand
Piecewise MILP under‐ and overestimators for global optimization of bilinear programs
Many practical problems of interest in chemical engineering and other fields can be formulated as bilinear programs (BLPs). For such problems, a local nonlinear programming solver often provides aExpand
...
1
2
3
4
5
...