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By Anil K. Jain

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The functions (2) and (3) are termed linear and nonlinear OHL networks, respectively. We borrow the term "OHL networks" from the parlance of neural networks, and we justify it by observing that the functions (3) have the same structure as feedforward neural networks with only one hidden layer and linear output activation units. The term OHL networks does not allow such a justification for the linear combination (2); however, we find it useful to join the functions (2) and (3) by a unifying terminology.

Let S be a subset of an infinite-dimensional real linear space H of vector functions I(;f) : B t-+ IRn2 , where B ~ IRnl. The functions I are the admissible solutions to the problem and F: S t-+ IR is a cost functional. The aim is to find an admissible solution that minimizes the cost functional. Even though the method described in the paper can be applied to functional optimization problems in general form, we focus on problems stated in stochastic environments. This will allow us to give special attention to theoretical and computational aspects resulting from the presence of random variables that, out of our control, are generated by the so-called state of the world.

35 Table 1, which shows different nonlinear ANs with rate 0 (lhfo) in some functional spaces, is a slight modification to Table 3 in [23, p. 255]. The function (J in entry 2 is sigmoidal. , Izl+ = 0 if z < 0, Izl+ = z if z ~ O. The function ~ in entry 4 has to satisfy a technical condition (see [26]); for instance, the squashing function, the generalized multiquadrics, the thin plate splines and the Gaussian function are allowed. d, and W;(K) is the Sobolev space of order s in the Lp(K) norm. 12)m/2, m > O.

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