Nonlinear programming: theory, algorithms, and applications
Nonlinear programming: theory, algorithms, and applications
Journal of Optimization Theory and Applications
On solving discontinuous extremal problems
Journal of Optimization Theory and Applications
The Minimization of Semicontinuous Functions: Mollifier Subgradients
SIAM Journal on Control and Optimization
Discontinuous piecewise linear optimization
Mathematical Programming: Series A and B
Algorithm 811: NDA: algorithms for nondifferentiable optimization
ACM Transactions on Mathematical Software (TOMS)
Optimization of Discontinuous Functions: A Generalized Theory of Differentiation
SIAM Journal on Optimization
Convexification and Global Optimization in Continuous And
Convexification and Global Optimization in Continuous And
Methods and Applications of Interval Analysis (SIAM Studies in Applied and Numerical Mathematics) (Siam Studies in Applied Mathematics, 2.)
McCormick-Based Relaxations of Algorithms
SIAM Journal on Optimization
Generalized McCormick relaxations
Journal of Global Optimization
Analysis of direct searches for discontinuous functions
Mathematical Programming: Series A and B
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A deterministic global optimization method is developed for a class of discontinuous functions. McCormick's method to obtain relaxations of nonconvex functions is extended to discontinuous factorable functions by representing a discontinuity with a step function. The properties of the relaxations are analyzed in detail; in particular, convergence of the relaxations to the function is established given some assumptions on the bounds derived from interval arithmetic. The obtained convex relaxations are used in a branch-and-bound scheme to formulate lower bounding problems. Furthermore, convergence of the branch-and-bound algorithm for discontinuous functions is analyzed and assumptions are derived to guarantee convergence. A key advantage of the proposed method over reformulating the discontinuous problem as a MINLP or MPEC is avoiding the increase in problem size that slows global optimization. Several numerical examples for the global optimization of functions with discontinuities are presented, including ones taken from process design and equipment sizing as well as discrete-time hybrid systems.