Main

## Main.ConstrainedOptimization History

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!!!! Interior Point Example

A refinery must produce 100 gallons of gasoline and 160 gallons of  diesel to meet customer demands. The refinery would like to minimize the cost of crude and two crude options exist. The less expensive crude costs $80 USD per barrel while a more expensive crude costs$95 USD per barrel. Each barrel of the less expensive crude produces 10 gallons of gasoline and 20 gallons of diesel. Each barrel of the more expensive crude produces 15 gallons of both gasoline and diesel. Find the number of barrels of each crude that will minimize the refinery cost while satisfying the customer demands.

* [[http://apmonitor.com/online/view_pass.php?f=refinery.apm | Solve Refinery Optimization Problem]] with Continuous Variables
* [[http://apmonitor.com/online/view_pass.php?f=irefinery.apm | Solve Refinery Optimization Problem]] with Integer Variables

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March 14, 2013, at 03:29 PM by 66.178.116.12 -
# A. Wächter and L. T. Biegler, On the Implementation of an Interior-Point Filter Line-Search Algorithm for Large-Scale Nonlinear Programming, Mathematical Programming 106(1), pp. 25-57, 2006. [[http://cepac.cheme.cmu.edu/pasilectures/biegler/ipopt.pdf|Download PDF]]
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December 22, 2012, at 09:28 AM by 69.169.188.188 -
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(:title Genetic Algorithms in Engineering Design:)
(:keywords genetic algorithms, continuous optimization, mathematical modeling, discrete optimization, nonlinear, optimization, engineering optimization, interior point, active set, differential, algebraic, modeling language, university course:)
(:description One often encounters problems in which design variables must be selected from among a set of discrete values
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(:title Constrained Optimization in Engineering Design:)
(:keywords constrained optimization, GRG, SQP, genetic algorithms, continuous optimization, mathematical modeling, discrete optimization, nonlinear, optimization, engineering optimization, interior point, active set, differential, algebraic, modeling language, university course:)
(:description Theoretical and numerical fundamentals of constrained optimization for engineering design
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December 22, 2012, at 09:25 AM by 69.169.188.188 -
(:title Genetic Algorithms in Engineering Design:)
(:keywords genetic algorithms, continuous optimization, mathematical modeling, discrete optimization, nonlinear, optimization, engineering optimization, interior point, active set, differential, algebraic, modeling language, university course:)
(:description One often encounters problems in which design variables must be selected from among a set of discrete values:)

[[Attach:chap6_constrained_opt1.pdf | Chapter 6: Constrained Optimization, Part I]]

We now begin our discussion of gradient-based constrained optimization. Recall that we looked at gradient-based unconstrained optimization and learned about the necessary and sufficient conditions for an unconstrained optimum, various search directions, conducting a line search, and quasi-Newton methods. We will build on that foundation as we extend the theory to problems with constraints.

At an unconstrained local optimum, there is no direction in which we can move to improve the objective function.  We can state the necessary conditions mathematically as Grad(f)=0. At a constrained local optimum, there is no feasible direction in which we can move to improve the objective.  That is, there may be directions from the current point that will improve the objective, but these directions point into infeasible space.

The necessary conditions for a constrained local optimum are called the Kuhn-Tucker Conditions, and these conditions play a very important role in constrained optimization theory and algorithm development.

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[[Attach:chap7_constrained_opt2.pdf | Chapter 7: Constrained Optimization, Part II]]

In the previous section we examined the necessary and sufficient conditions for a constrained optimum. We did not, however, discuss any algorithms for constrained optimization. The purpose of this section is to review three popular techniques for constrained optimization: