Physics 823 on Optimization Techniques

Summary:

The PHY 823 course on Optimization Techniques focuses on understanding optimization through Taylor’s series approximation and the principle of minimum potential energy. The course discusses the second-order Taylor’s series approximation of functions. The second-order approximation was provided using a series of mathematical equations for a given function. The course then delves into necessary and sufficient conditions for determining extreme points, leveraging the Hessian matrix for this purpose. An example showcased involves a function, its partial derivatives, and the Hessian matrix to ascertain the nature of its extreme points. Another example examines two frictionless rigid bodies connected by springs and uses the principle of minimum potential energy to find their displacements under a force. The concept of a saddle point, a point where the Hessian matrix may be neither positive nor negative definite, is introduced. A saddle point has the characteristic of a relative minimum or maximum with respect to one variable while being the opposite for the other.

Excerpt:

Physics 823 on Optimization Techniques

COURSE CODE: PHY 823

COURSE TITLE: OPTIMIZATION TECHNIQUES

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Solution by Direct Substitution

For a problem with n variables and m equality constraints:
• Solve the m equality constraints and express any set of m variables in terms of the remaining n-m variables
• Substitute these expressions into the original objective function; the result is a new objective function involving only n-m variables
• The new objective function is not subjected to any constraint; hence, its optimum can be found using unconstrained optimization techniques.
• Simple in theory
• Not convenient from a practical point of view as the constraint equations will be nonlinear for most of the problems
• Suitable only for simple problems

Example 5: Find the dimensions of a box of the largest volume that can be inscribed in a sphere of unit radius
Solution: Let the origin of the Cartesian coordinate system x1, x2, x3 be at the sphere’s centre, and the box’s sides be 2×1, 2×2, and 2×3. The volume of the box is given by:

1 2 3 1 2 3 f (x , x , x ) = 8x x x