Optimization techniques primer

The Data Collaboration approach casts problems as constrained optimization over the feasible region, drawn on the entire knowledge 

All the questions posed in the Data Collaboration framework appear as constrained optimization problems [72]. Typically, there are both inequality and equality constraints. If /, g, and h are functions, then a constrained optimization problem is of the form 

The variable x is the optimization variable. The set of values of x which satisfy the two constraints is called the feasible set. Replacing / with —/ recasts the problem from minimization to maximization. Without loss of generality we restrict the discussion to minimizations. 

Ideally, the solution to an optimization problem consists of an optimum value, Xopt, for the optimization variable and a proof (often called the certificate) of its optimahty. Alternatively, one may obtain a subop-timal point, Xsopt, that is feasible and achieves an objective value /(Xsopt) within some fixed tolerance of the optimum. For general optimization problems such as those in equation (20) this preciseness is rarely obtained, usually requiring a form of optimization called global optimization. 

Many algorithms can be shown [72] to generate a convergent sequence [Xitl i whose limit point is a local optimum of (20). A point x is a local optimum if it satisfies the constraints and there is an open ball containing X such thaty(x) /(x ) for every point x in the open ball satisfying the constraints. Under suitable hypotheses, algorithms such as gradient descent and Newton s method produce such sequences. 

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We have also used mutagenic primers to insert histidine tags and epitope tags into cDNAs of interest (Neish et al, 2003). Inserting several codons into a cDNA is a little more difficult than simply substituting one amino acid for another, but is quite feasible using this technique. Optimization of the reaction is absolutely critical when performing more difficult manipulations such as this. 

An isothermal nncleic acid sequence-based amplification (NASBA) assay was optimized to amplify viral RNA of all four dengue virus serotypes by a set of universal primers and to type the amplified products by serotype-specific capture probes. The NASBA assay involved the use of silica to extract viral nucleic acid, which was amplified without thermocycling (Wu et al. 2001a). The main advantage of this assay over a PCR technique is that it is entirely isothermal and is conducted at 41°C. Thus, it would be suitable for epidemiological studies in the field (Wu et al. 2001a). 

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