Question

Discuss the differences between linear and nonlinear optimization methods. Include specific examples of nonlinear optimization methods, and discuss which aspects of these models make them different from traditional linear optimization methods.

Answer 1

Answer:

Given Data

An optimization model is defined by an objective function dependent on a set of variables (i.ee ., decision variables ) restricted by various constraints . The goal of optimization is to find a set of decision variables that generates the optimal value for the objective function while satisfying all the imposed constraints.

Linear optimization model is a mathematical model fpr determining a way to achieve the best outcome ( such as maximization profit or lowest cost ) for some list of requirements represented as linear relationships.

Nonlinear optimization model is the mathematical model of solving a system of equalities and inequalities , collectively termed constraints , over a set of unknown real variables , along with an objective function to be maximised or minimized , where some of the constraints or the objective function are nonlinear.

Portfolio Selection : An investor has $5000 and two potential investments. Let for j = 1 and j = 2 denote his allocation to investment j in thousands of dollars . From historical data , investments 1 and 2 have an expected annual return of 20 and 16 percent , respectively . Also , the total risk involved with investments 1 and 2 , as measured by the variance of total return, is given by , so that risk increases with total investment and with the amount of each individual investment. The investor would like to maximize his expected return and at the same time minimize his risk. Clearly , both of these objectives cannot , in general , be satisfied simultaneously. There are several possible approaches . For example , he can minimize risk subject to a constraint imposing a lower bound on expected return. Alternatively , expected return and risk can be combined in an objective function , to give the model:

Subject to :

The nonnegative constant reflects his tradeoff between risk and return.

If = 0 , the model is a linear program , and he will invest completely in the investment with greatest expected return.

For very large , the objective contribution due to expected return becomes negligible and he is essentially minimizing his risk.

The electric power sysytem is a wide geographical distribution of network , which is associated with linear and non-linear mathematical relations. Earlier , these problems are solved using tradition optimization techniques such as gradient methods ,linear and non-linear programming , quadratic programming , newton method , P-Q decomposition and interior point method . recent power system problems are associated with multiobjective and real time constraints , with more than one local optimal solution.

Therefore , these optimization techniques are not suitable for such complex problems , because they may not be able to provide a global optimum solution.

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