8-pCPT-2-O-Me-cAMP-AM We consider the following perturbation

We consider the following perturbation of problem (2):where ∊>0 is an identity element. For a given ∊, we solve problem (3) to get the solution (x, y, λ). As ∊ decreases slowly toward 0, the optimum of F follows to the global optimum of F0. This set of constraint conditions is called the central path conditions as they 8-pCPT-2-O-Me-cAMP-AM define a trajectory approaching the solution set for ∊→0. The smoothing method is an effective way to deal with this issue. The main feature of smoothing method is to reformulate the problem (3) as a smooth nonlinear programming problem by replacing the difficult equilibrium constraint with a smoothing function. Consequently, the true problem (3) can be approximated by a sequence of smooth problem [21], [28], [29]. The numerical evidence on the efficiency of the smoothing approach is very impressive. Hence, In this work, we smooth the nonlinear programming with complementary constraints by applying CHKS smoothing function. Define function ϕ: R3→R using the CHKS smoothing function . Then, we can have the following proposition [21]. Note that for every ∊⩾0, ϕ(a, b) is continuously differentiable with respect to (a, b). By applying the CHKS smoothing function ϕ(x, y), problem (3) can be approximated by: Problem (4) can be viewed as a perturbation of problem (2). Problem (4) overcomes the difficulty due to problem (2) not satisfying any regularity assumption and paves the way for using an augmented Lagrangian multiplier approach to solve problem (2). The idea of the algorithm is as followed: the smoothing method starts with a large ∊, finds the optimum of F, decreases ∊ gradually, and recomputes the optimum at each step-starting from the previous solution-until ∊ is small enough. We claim that this procedure yields near-optimal solutions. To simply our discussion and for convenience, we introduce the following notations: Let h(x, y, λ)=G(x, y), and z=(x, y, λ), then problem (4) can be written as: Similar to the main result, Theorem 3, in [21], the theorem below follows directly.
Augmented Lagrangian multiplier method
Numerical experiments In this section, we present three examples to illustrate the feasibility of the augmented Lagrangian multiplier algorithm for NBLP problems.
Conclusions
Acknowledgements
Introduction Receptor-ligand interactions or in vitro stimulation of monocytes with mitogens both initiate cell surface signal transduction pathways that lead to gene transcription and cellular activation. In many cases, activation of receptor-associated protein tyrosine kinases (PTKs) mediate these effects via phosphorylation of key cellular substrates. The src-family of non-receptor PTKs represents the largest known family of PTKs. To date, nine members which share protein domain structure have been identified (Src, Yes, Fyn, Lyn, Lck, Blk, Hck, Fgr, and Yrk) (Korade-Mirnics and Corey, 2000). These proteins participate in many of the critical activation steps for cellular functions (Korade-Mirnics and Corey, 2000), and for this reason, regulation of these kinases is crucial in the regulation of immunity. In macrophages, only certain members of the Src family are expressed, primarily Hck, Fgr, Lyn, Yes, and Fyn (Korade-Mirnics and Corey, 2000). The observation that differentiation of macrophages specifically induces Hck expression while leaving Lyn levels unchanged and reducing Fgr expression (Boulet et al., 1992, Ziegler et al., 1988), suggests the need for tightly regulated levels of Src-family kinase activity at various stages of activation and differentiation. Furthermore, the increase in phosphorylation of various Src-family members following phagocytosis (Wang et al., 1994, Hamada et al., 1993, Ghazizadeh et al., 1994), engagement of MHC class II antigen (Morio et al., 1994), stimulation with IL-3 or GM-CSF (Jucker and Feldman, 1995), or the activation of macrophages with LPS or IFN-γ (Boulet et al., 1992, Ziegler et al., 1988) indicates that these PTK serve many functions in signal transduction during monocyte activation. In addition, the association of several different Src-family kinases with the same cell surface receptor (Thomas and Brugge, 1997) suggests redundancy in this aspect of monocyte signaling, and demonstrates the importance of these PTKs in host defense.