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...can in principle be solved efficiently by an interior point method. To conclude, we consider two more applied nonconvex examples. The first is due to Mangasarian [23] and is used by Jokar and Pfetsch =-=[18]-=- to find sparse solutions of underdetermined linear equations. The formulation of [18] can be stated in the form (2.3) where the regularization function | · |∗ has the form |x|∗ = n∑ (1 − e −α|xi| ) i...

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...number of nonzero reaction rate coefficients [72]. It is remarked that the computation of sparse realizations is an NP-hard problem, where generally mixed integer linear programming cannot be avoided =-=[82]-=-. There exist certain conditions under which the problem can be solved in polynomial time [83] but these are often not fulfilled in the case of CRNs. Moreover, there are effective heuristics to addres...

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...ΦS)i,j|. The mutual incoherences M k are intimately related to the best constant δ k with which the matrix Φ satisfies RIP of order k, but a full understanding of this connection has not been reached =-=[77, 78]-=-. A challenging aspect of RIP is its computational cost. Indeed, RIP is a property of the submatrices of a specific size. At present, no subexponential-time algorithm is known for testing RIP. Introdu...

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...ient conditions that guarantee that an (approximate) sparse solution ˆx to (1) can be obtained by solving the linear program min ‖x‖1, s.t. Φx = b (‖Φx − b‖ ≤ ǫ), which can be done in polynomial time =-=[30, 31, 32]-=-. We will give a brief survey of this theory in Section 2.2. In the literature, the development has mostly focused on the construction of appropriate coding matrices Φ that allow for the sparse repres...

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... used a similar formulation to derive a diagnosis tool robust against sensor faults, see [11]. A further step to enhanced fault isolation can be taken by recognizing that abrupt events involve only a limited number of health parameters, which translates mathematically into the search for a sparse solution. Basically, the regularisation term is again tweaked in order to favour a solution with many nought terms. Problems characterised by sparsity are encountered in many scientific fields such as compressed sensing [12], linear regression [13] or source localisation [14] to name a few. Reference [15] provides a good review of various algorithms looking for sparse solutions. Moreover, it is shown in [16] that one flavour of the sparse estimation problem can be cast as a Quadratic Programming problem for which efficient solvers are available. In the present contribution, this Quadratic Programming approach is adopted to derive a fault isolation tool favouring sparse solutions. As a validation procedure, the resulting algorithm is applied to a variety of fault conditions simulated with a generic commercial turbofan model. DESCRIPTION OF THE METHOD The scope of this section is to provide the ...

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...number of nonzero reaction rate coefficients [72]. It is remarked that the computation of sparse realizations is an NP-hard problem, where generally mixed integer linear programming cannot be avoided =-=[82]-=-. There exist certain conditions under which the problem canbesolvedinpolynomialtime[83]buttheseare often not fulfilled in the case of CRNs. Moreover, there are effective heuristics to address the pro...

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...closely related problem to the cardinality constrained quadratic program is the ℓ0-norm minimization problem or sparse solutions of linear equations: min { ‖x‖0 | Ax = b } . The reader is referred to =-=[10, 35, 47]-=- for an extensive literature on this problem. In this section, we describe different inexact methods for (Pc) or its special cases. These methods are mainly based on various approximations and relaxat...