AOE/ESM 4084:
Engineering Design Optimization
and
Multidisciplinary Design Optimization (MDO)

Course Description (PDF) :
This is an elective course for seniors in AOE and ESM departments. The course is also approved for graduate credit, and is typically taken by beginning graduate students as a prerequisite course for the AOE/ESM 5064 "Structural Optimization".

• Lectures
• Tutorials
• Homeworks
• Final Project
• Programs

Lectures:
Following are some of the viewgraphs that are used in classroom. The links provided below are PDF files.

• Introduction to design and optimization.
  Lecture-01 :
  
  
• Mathematical formulation of design optimization problems.
  Lecture-02 :
  
  
• Graphical solution of optimization problems.
  Lecture-03 :
  
  
• Multicriteria optimization.
  Lecture-04 :
  
  
• Fundamental concepts of optimality-I. Gradient vector, Hessian matrix, Taylor series expansion, Quadratic forms and Eigenvalues of matrices.
  Lecture-05 :
  
  
• Fundamental concepts of optimality-II. Necessary and sufficient conditions for optimality of unconstrained and equality constrained problems.
  Lecture-06 :
  
  
• Fundamental concepts of optimality-III. Necessary and sufficient conditions for optimality of constrained problems, Kuhn-Tucker conditions.
  Lecture-07 :
  
  
• Fundamental concepts of optimality-IV. Global optimality, convex functions, convex programming problems, and post optimality analysis.
  Lecture-08 :
  
  
• Linear Programming (LP), and Sequential Linear Programming (SLP) with move limits implementation.
  
  
• One dimensional minimization; Bracketing, polynomial interpolation and Golden section search.
  Lecture-09 :
  
  
• REVIEW - I :
• Test - I:
• Solution of Test I:
  
  
• Unconstrained Minimization: Nelder and Mead's Sequential Simplex, Powell's Conjugate Directions, Steepest Descent, Fletcher-Reeves Conjugate Gradient, Newton's, and BFGS Quasi-Newton Algorithms.
  
  
• Sequential Unconstrained Minimization Techniques (SUMT) for n-dimensional constrained function minimization; Exterior, Interior, and Extended Interior Penalty function approaches.
  Lecture-11 :
  
  
• Method of Multipliers (Augmented Lagrange Multiplier Method) for equality and inequality constrained problems.
  Lecture-12 :
  
  
• REVIEW - II :
  
  
• Method of Feasible Directions.
  Lecture-13 :
  

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Tutorials:
Following is a list of Mathematica tutorials that can be used to supplement lectures througout the semester. Students are encouraged to go through these tutorials after each lecture and try their own variations of the examples.
NOTE: For those who have Mathematica installed on their machines, clicking any of the .ma (Mathematica Version 2) or .nb (Mathematica Version 3) links may enable you start up the local Mathematica program automatically. If it doesn't see your system administrator or web browser documentation for adding Mathematica to your browser's helper applications.

• This tutorial is to demonstrate the use of Mathematica and Mathematica graphics as a classroom teaching and tutorial environment.
  Tutorial-01.nb or Tutorial-01.ma :
  
  
• Mathematical formulation of design optimization problems. Definition of design variables, onjective functions, and constraints in terms of Mathematica lists and functions.
  Tutorial-02.nb or Tutorial-02.ma :
  
  
• Graphical solution of 2-D optimization problems using Mathematica graphics, and Mathematica Solve function.
  Tutorial-03.nb or Tutorial-03.ma :
  
  
• Example of a two objective optimization problem, and Pareto optimal curve.
  Tutorial-04.nb or Tutorial-04.ma :
  
  
• Fundamental concepts of optimality. Taylor series expansion, Gradient vector, Hessian matrix, Quadratic forms, and Eigenvalues of matrices.
  Tutorial-05.nb or Tutorial-05.ma :
  
  
• Fundamental concepts of optimality-II. Necessary and sufficient conditions for optimality of unconstrained and equality constrained problems.
  Tutorial-06.nb or Tutorial-06.ma :
  
  
• Fundamental concepts of optimality-III. Necessary and sufficient conditions for inequality constrained problems (Kuhn-Tucker conditions). Post optimality analysis.
  Tutorial-07.nb or Tutorial-07.ma :
  
  
• Linear Programming (LP) and linearization of nonlinear functions.
  Tutorial-08.nb or Tutorial-08.ma :
  
  
• Sequential Linear Programming (SLP) example.
  Tutorial-09.nb or Tutorial-09.ma :
  
  
• Implementation of move limits strategy and use of a Mathematica program for Sequential Linear Programming, SequentialLP.m (see Programs to obtain a copy).
  Tutorial-10.nb or Tutorial-10.ma :
  
  

• One-dimensional unconstrained minimization. Conversion of n-D problems to 1-D problems, bracketing the minimum, polynomial approximation, golden section search, and the use of built-in Mathematica functions.
  Tutorial-11.nb or Tutorial-11.ma :
  
  

• Demonstration of Nelder and Mead's Sequential Simplex algorithm for unconstrained minimization of n-dimensional functions. A Mathematica function called SequentialSimplex is included in the package UnconstrainedMin.m (see Programs to obtain a copy).
  Tutorial-12.nb or Tutorial-12.ma :
  
  
• Powell's conjugate directions method for unconstrained minimization of n-dimensional functions. A Mathematica function ConjugateDirections is included in the package UnconstrainedMin.m (see Programs to obtain a copy).
  Tutorial-13.nb or Tutorial-13.ma :
  
  
• Steepest descent algorithm. A Mathematica function SteepestDescent is included in the package UnconstrainedMin.m (see Programs to obtain a copy).
  Tutorial-14.nb or Tutorial-14.ma :
  
  
• Fletcher-Reeves conjugate gradient algorithm. A Mathematica function ConjugateGradient is included in the package UnconstrainedMin.m (see Programs to obtain a copy).
  Tutorial-15.nb or Tutorial-15.ma :
  
  
• Newton's method for unconstrained minimization of n-dimensional functions. A Mathematica function ConjugateGradient is included in the package UnconstrainedMin.m (see Programs to obtain a copy).
  Tutorial-16.nb or Tutorial-16.ma :
  
  
• Quasi-Newton methods for unconstrained minimization of n-dimensional functions. A Mathematica function QuasiNewton is included in the package UnconstrainedMin.m (see Programs to obtain a copy).
  Tutorial-17.nb or Tutorial-17.ma :
  
  
• Penalty function approach for constrained minimization of n-dimensional functions. A Mathematica function PenaltyFunction is included in the package ConstrainedMin.m (see Programs to obtain a copy).
  Tutorial-18.nb or Tutorial-18.ma :
  
  
• Augmented Lagrange multiplier method and its demonstration for the equality constrained problems and inequality constrained problems.
  Tutorial-19.nb or Tutorial-19.ma :
  
  
• The method of Feasible Directions, and demonstration of the push-off factor.
  Tutorial-20.nb or Tutorial-20.ma :
  
  
•  

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Homeworks:
Following are the statements of the homework assignments. The links for the assignments are PDF files. Solutions are Mathematica notebooks.

• Ploting of the cargo weight as a function of aspect ratio, wing areea, Mach number, and sweep angle using Mathematica.
  HW-01: Analysis Hint: Solution:
  
  

• Graphical solution of a design optimization problem
  HW-02 : Solution:
  
  

• Multiobjective design of a beam cross section.
  HW-02 : Solution:
  
  
• Taylor series approximations of a function.
  HW-03 : Solution:
  
  
• Graphical solution of the cargo weight maximization as a function of the Mach number and the wing area.
  HW-05 : Solution:
  
  
• Solution of Kuhn-Tucker conditions to find stationary points, and post optimality analysis.
  HW-04 : Solution:
  
  
• Proof of post optimality analysis (mandatory for graduate students).
  HW-Extra-Credit-1 : Solution:
  
  
• Formulation and solution of a Linear Programming problem.
  HW-05 : Solution:
  
  
• Maximization of cargo weight with a lift constraint as a function of the Mach number and the wing area using Sequential Linear Programming.
  HW-06 : Solution:
  
  
• Minimization of an unconstrained function using Sequential Simplex algorithm.
  HW-07 : Solution:
  
  
• HW-Extra-Credit-2 :
  Computation of the Steepest Descent direction. (Mandatory for graduate students)
  
  

• Maximization of cargo weight as a function of the Mach number and the wing area using derivative based unconstrained minimization algorithms.
  HW-08 : Solution:
  
  

• HW-Extra-Credit-3 :
  Computation of the Fletcher Reeves Conjugate Gradient Direction. (Mandatory for graduate students)
  
  

• Maximization of cargo weight as a function of the Mach number and the wing area using the Augmented Lagrange Multiplier Method
  HW-9 : Solution:
  
  

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Project:
Following is the description of the final course project. Each student will be assigned with a different set of load magnitudes and displacement constraints.

• Final Project : Stresses:

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Programs:
Following are some of the Mathematica programs that will be used in the course. They need to be copied to a disk on your machine and loaded ( << ProgramName.m ) into the notebook.

• SequentialLP.m :
  
• UnconstrainedMin.m :
  
• ConstrainedMin.m :
  
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