As we will not obtain any improvements along that

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Unformatted text preview: n will also be greater than f (z * ) . As we will not obtain any improvements along that particular branch we can prune it (i.e. get rid of it) The branching and the pruning are the two basic components in branch and bound. Implementations differ in how the branching components are selected.12 In a worst-case situation we might, however, end up solving all of the subproblems RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 46 Practical Considerations When Using Optimization Software We have to be careful when the use numerical routines as “black boxes” Why? The incorrect usage of numerical software may lead to reduced efficiency, lack of robustness and loss in accuracy RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 47 The Solution Process The solution process for solving an optimization problem can be divided into three parts: Formulating the problem Choosing an optimizer Solving the problem with the optimizer RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 48 Formulating the Problem Numerical optimization solvers require that the problem is stated in a standard form: The first step in solving an optimization problem with numerical software is to identify its type Sometimes this is straightforward as the problem might already be given in some standard form o However, more often than not this is not the case and the original problem has to be transformed into one of the standard forms RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 49 Choosing an Optimizer When it comes to the choice of optimization algorithms, unfortunately, there is no single technique that is better or outperforms all the others Unrealistic to expect to find one software package that will solve all optimization problems Different approaches and software packages are often complementary and some are better suited for some problems than others In practice I recommended you try different algorithms on the same problem to see which one performs best as far as speed, accuracy, and stability Although it is possible to solve a simple linear program with the nonlinear programming algorithm, this is not necessarily advisable. In general, we can expect more specialized algorithms to solve the problem not just faster but also more accurately RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 50 Constraints Whether a problem is constrained or unconstrained will affect the choice of algorithm or technique that is used for its solution In general, unconstrained optimization is somewhat simpler than constrained optimization The type of constraints also matter. Problems with equality constraints are in general easier to deal with than inequality constraints, as is linear compared to nonlinear constraints RISK AND PORTFOLIO MANAGEMENT WITH ECONOMETRICS, VER. 11/21/2012. © P. KOLM. 51 Derivatives Many optimization routines use derivative information It is great if some or all of the first order derivatives (and sometimes also second-order derivatives) of the objective function and constraints are available analytically If not, but all the functions involved are differentiable, then the algorithm will have to calculate these derivatives numerically As a general rule of thumb, if analytic derivatives can be supplied by the user this will greatly speed-up each iteration...
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