LECTURE NOTES Non-Linear Programming

Transcription

1 CEE 6110 David Rosenberg p. 1 Learning Objectives LECTURE NOTES Non-Linear Programming 1. Write out the non-linear model formulation 2. Describe the difficulties of solving a non-linear programming model compared to a linear programming model 3. List and differentiate the based solution methods to solve nonlinear optimization problems 4. Use a piecewise linearization method to solve a convex nonlinear optimization problem 5. Apply gradient based (steepest ascent) method to solve a nonlinear optimization problem 6. Solve nonlinear programming problems using Excel and GAMS 1. Non-Linear Model Formulation Min (or Max) Z = f(x1, x2,, xn) Subject to: x1, x2,, xn = f = g1, g2,, gm = Can include: Objective function and all constraints non-linear A non-linear objective function with linear A linear objective function with nonlinear Linear and non-linear constraints

2 CEE 6110 David Rosenberg p Difficulties in Solving Non-Linear Problems Problem Aspect Linear Problems Non-Linear Problems Solution space shape Location of optimal solution Solution Type All starting points lead to the same solution Solution outcomes Global Feasible model=>global optimum Feasible model, unbounded Infeasible model Additional Challenges for Non-Linear Problems: Difficult to satisfy equity constraints (and keep them satisfied) Complex math theory and algorithms. Which algorithm to apply and can you implement the steps correctly? Difficult to determine whether your problem formulation satisfies a solver s requirements. Different algorithms and solvers arrive at different solutions for the same formulation. Different but equivalent formulations of the model produce different solutions. Difficult to use nonlinear solvers. Numerous parameters and difficult to understand how the parameters affect the solution. Some Recommended Best Practices to Get Around the Challenges 1. Use the simplest model formulation possible. 2. Know the characteristics of your model before choosing a solution algorithm. 3. Provide a good starting point. Better to choose closer to the optimum. 4. Put reasonable bounds on all variables. Avoid searching unnecessary parts of the feasible region. 5. Always use a modeling language (e.g., GAMS). Allows easy switch between solvers (when one does not work).

3 CEE 6110 David Rosenberg p Non-Linear Solution Strategies Method Approach Requirements 1) Classical (analytical; calculus) 2) Reduce to linear form (piecewise linearization) Calculate 1 st /2 nd derivatives Continuous variables Satisfy necessary / sufficient K-T conditions Solution space is a convex set Variables are separable 3) Numerical search i. Gradient based Steepest ascent, conjugate gradient, reduced gradient ii. Direct search Genetic algorithms, simulated annealing, cutting planes, branch and bound Varies by method QUESTION: What is the benefit to developing and solving nonlinear problems if a global optimal solution is not guaranteed? 4. Use Piecewise Linearization to Solve Example 1. What is the solution to problem from Bishop et al.? Solution: See Ex8-4-1.xls

4 CEE 6110 David Rosenberg p Apply Gradient-Based Methods to Solve Basic Steps i. Select a (good) starting point ii. Determine direction to next point (gradient) iii. Determine step side iv. Evaluate convergence criteria (stopping rule) o Number of iterations o Change from previous solution v. Investigate optimum o Is it global? (if not, you are stuck and don t even know it o What then? Example 2. Find the solution to Example using a gradient based method. Solution:

5 CEE 6110 David Rosenberg p Use Excel and GAMS to Solve In Excel Specify non-linear functions in cells Select Solving Method as GRG Nonlinear Solve the model Check the solution report stop criteria. Example 3. Find the solution to Example using Excel s Generalized Reduced Gradient solver. How do the solution reports differ from the reports for the solution to an LP problem? Answer: In GAMS Enter equations as nonlinear functions of the decision variables and parameters. Solve the model as an NLP Check the solution report and solver and model statuses An example solution is provided in Ex8-4-1.gms Wrap Up 1. Non-linear problems are much more difficult to solve 2. Three overall solution strategies -- analytical, piecewise linearization, and numerical methods 3. Numerical methods do not guarantee an optimal (truly best) solution for most non-linear programs 4. Lots of software to use choice is often problem and problem formulation specific