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Linear Programming and Mixed Integer Programming
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Continuous constraints produce infinite dimensional optimization problems, unlike all the LP problems previously discussed in this chapter, which have a finite number of constraints. Consider the constraint f (ω) ≤ 0, ∀ω ∈ Ω, where Ω is an uncountable set, for example an interval [α, β]; for the sake of simplicity, we consider a scalar function, depending on a single parameter, ω in our case. Assume that the function has the form () f(ω)=ϕ0(ω)+∑k=1Nxkϕk(ω),
On the implementation of a quasi-Newton interior-point method for PDE-constrained optimization using finite element discretizations
Published in Optimization Methods and Software, 2023
Cosmin G. Petra, Miguel Salazar De Troya, Noemi Petra, Youngsoo Choi, Geoffrey M. Oxberry, Daniel Tortorelli
A reader with background in nonlinear programming (NLP) in finite-dimensional Euclidean spaces may disagree with our choice of considering a complicating infinite-dimensional setup. Since the infinite-dimensional optimization problems we consider are invariably discretized, one may suggest instead to pose the discretizations as ‘finite-dimensional’ problems and to solve them using NLP solvers over the Euclidean space. However, using the Euclidean inner product instead of the inner product of the underlying Hilbert space causes incorrect representers to be used for certain derivative functionals and introduces discretization inconsistencies (e.g. mesh dependence) in the discretized optimality conditions. These discrepancies cause convergence behaviour dependent on the underlying discretization or meshing of the domain (see [49] and Section 3.6.1 for examples of simple problems and Section 6 for more complex problems where this issue is pervasive). Such mesh dependent behaviour is not necessarily specific to optimization and has been previously identified and addressed in the context of linear systems arising in PDE discretizations (for example, see [24,30]).