This website uses cookies to ensure you have the best experience.

# Simulation Essay

1260 words - 6 pages

Steepest Descent Direction in Optimization - Application and Algorithm

Hemanand. T
Department of Chemical Engineering,
St. Joseph’s College of Engineering,
Chennai – 600 119

Abstract: An analytical solution to identify the minimum value of a function used for optimization based on steepest descent technique was extensively discussed with applications in a process. The properties of gradient vector, the oscillation of function values and overshoot were analyzed in a function for the search of minimum. The best step size in each iteration was found by conducting a one-D optimization in the steepest descent direction. The five steps in the algorithm for steepest descent direction ...view middle of the document...

The first optimization technique, which is known as steepest descent, goes back to Gauss. Historically, the first term to be introduced was linear programming, which was invented by George Dantzig in the 1940s.
Gradient descent is a first-order optimization algorithm. To find a local minimum of a function using gradient descent, one takes steps proportional to the negative of the gradient (or the approximate gradient) of the function at the current point. If instead one takes steps proportional to the gradient, one approaches a local maximum of that function; the procedure is then known as gradient ascent. Gradient descent is also known as steepest descent, or the method of steepest descent. When known as the latter, gradient descent should not be confused with the method of steepest descent for approximating integrals.

2. Properties of steepest descent technique:
Properties of Gradient Vector
The gradient vector of a scalar function [pic]is defined as a column vector
[pic]

For example

[pic]

at the point[pic]

[pic]

The normalized gradient vector
[pic]

For example, at the point [pic]

[pic]

2.1 Property 1. The gradient vector represents a direction of maximum rate of increase for the function f(x) at x*. For example,

[pic]

If we increase x in the direction [pic] by a step size of α = 0.5
[pic]
The function value becomes
[pic]

If we move in a direction [pic]

[pic] + .5[pic] [pic]

The function value becomes

[pic]

If we move in a direction [pic]

[pic] + .5[pic] [pic]

The function value becomes

[pic]

We can see that moving along the gradient direction results in direction results in the maximum increase in the function.

2.2 Property 2.

The gradient vector c of [pic]) at the point [pic]is orthogonal (normal) to the tangent plane for the surface [pic] constant. For example, [pic])= 25[pic] + [pic] = 25 the slope at [pic]=.6, [pic]=4 can be found to be

2(25)[pic]

Slope = [pic] = -[pic]

The direction of the tangent line is given by

t = [pic][pic]

c and t are normal each other as

[pic]t = [pic] = 30-8(3.75)=0

2.3 Property 3:

The maximum rate of change of [pic] at any point [pic] is the magnitude of the gradient vector by

[pic]

Steepest descent direction. Let [pic] be a differentiable function with respect to X. The direction of steepest descent for [pic] at any point is

d=-c or [pic]

3. Application of steepest descent direction:

3.1 Application of the steepest descent direction to search for the minimum for [pic]=25[pic] + [pic] starting at [pic]=[pic] with a step size of α=.5. the function value at the starting point is
[pic]

Ana analytical solution reveals that the minimum point is at [pic]=[pic] and[pic]. Let us start the process of iterations.

[pic]

[pic] - .5[pic] [pic]

[pic]

[pic]

[pic] - .5[pic] [pic]

[pic]

[pic]

[pic] - .5[pic] [pic]
...

## Other assignments on Simulation

### Marketing Plan Essay

1924 words - 8 pages Executive Summary The purpose of writing this report is to evaluate and provide sound advices to the company, Venture Simulation Ltd on its business simulation game, Sim Venture. Several recommendations will be given for future improvements based on the main business areas of the company. In recent years, the business simulation industry is growing rapidly. With outstanding product design and feature, more and more educational institutes

### Hnee 115 Ecad Essay

2165 words - 9 pages Amplifier Simulation 1 2 Op Amp Summing Amplifier Simulation 2 3 Results Simulation 1 & 2 5 2) Describe the following terms used in ECAD systems: 6 (a) Prototype 6 (b) Boundary scan 6 (c) Self-testing 7 (d) Controllability and observability 7 (e) Testability 7 (f) Simulation modelling 7 (g) Fault simulation 8 (h) Temperature simulation 8 3) VLSI 9 Conclusion 10 References 10 Online sources 10 Books 11 Assignment

### Individual Reflections

3450 words - 14 pages Table of Contents Page 1.0 Introduction ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙ 1 2.0 The Group Selection ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙ 1 3.0 SimVenture Simulation ∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙∙ 2 4.0 Experience with SimVenture

### Termpaper

801 words - 4 pages Supply and Demand Simulation ECO/365 July 31, 2013 Robert Watson Supply and Demand Simulation This paper is about the supply and demand simulation from University of Phoenix student website. The simulation is about Good life Management, a real estate property management company that controls the seven apartment complexes in the city of Atlantis. In the nine-year period in the simulation the housing market has increased and decreased

### Buisness

385 words - 2 pages Student Name: _________________________ Facilitator: _________________________ Complete the interactive Ethics Game simulation, located on the course page on the University of Phoenix Student Web site for Week Three. You will be able to go through the simulation only one time, so please use this worksheet to take notes. The simulation will use the following decision model with the indicated critical decision points. You will want to take

### Homework

486 words - 2 pages sell off the capacity of Baker and kill this product to gain sales and end this simulation in a positive form. For this round, Baldwin team decided to decrease the promotional budget for Baker and Buddy to save money since it already reached a 100% awareness. In addition, the team members decided to reduce the products' prices in order to sell more units and get rid of what is leftover in the inventory. Even though the team members were

### Management

858 words - 4 pages Achieving Project Goals Simulation Maria Algarin University of Phoenix MGT 437 Project Management Simulation It is important for project managers to not only make the best project plans, but to be also suitably prepared to tackle contingencies and minimize their impact on the project (Achieving Project Goals Simulation, 20011). In my opinion, there is a misconception as to the role of a project manager. The main challenge is to

### Eth 316 Week 5

953 words - 4 pages Ethics Game Dilemma ETH 316 Ethics Game Dilemma Various conditions come up in the office that requires employees to sit down and think about the best solution. This usually involves supervisors and subordinates. At times answers are vague and what seems to be the most clearly defined answer, is not always the suitable choice. The Ethics Game simulation displayed two different cases: The Case of the Mysterious Rose and The Case of the Cold

### Hcs 440 Week 5 Learning Team Health Care Reform Project Part Iii

374 words - 2 pages This paperwork contains HCS 440 Week 4 Individual - Economic Issues Simulation Paper General Questions - General General Questions Review the Health Care Economics Issues HMO Simulation exercise posted on your student web page. Follow the Wk 4 WEB LINK “Understanding Economic Issues for HMO’s”. This exercise is for your learning experience only. Do not post any screen short from the exercise as part of your assignment

### Xyz Company

651 words - 3 pages Summary: XYZ is concerned about the plant’s present production capability. The Vice President, Katherine Rally is contemplating three alternatives: 1) Full automation; 2) Semi-automatic; and 3) Doing nothing. She identified Capital investment, Annual Revenues, Useful life, and Salvage Value of the plants as random variables and noted down their respective probability distributions. The case requires developing a simulation of 50 sample

### Improving Organizational Performance

1318 words - 6 pages Week 3 Individual Assignment “Improving Organizational Performance” Simulation Summary Diana Salisbury PSY/428 November 8, 2010 University of Phoenix Adinah Johnson CERTIFICATE OF ORIGINALITY: I certify that the attached paper, which was produced for the class identified above, is my original work and has not previously been submitted by me or by anyone else for any class.  I further declare that I have cited all sources from which I used

## Similar Documents

### Week 4 Simulation Relation Essay

489 words - 2 pages University of Phoenix Material Simulation Review Complete a simulation and then write a paper that analyzes the decisions you made. You are in charge of fictitious health care organization facing financial difficulties. In the simulation, you will complete the following tasks: Bridge a working capital shortage. Evaluate funding options for acquiring medical equipment. Evaluate funding options for capital expansion. Part I below

### International Legal And Ethical Issues Simulation Summary

590 words - 3 pages An international company should have a very specific contract agreement with their out of country partners specifying the form of law as well as the forum of which disputes are handled. Some of the issues addressed in this simulation were ethical issues and government issues. One ethical issue was when the country was having a viral epidemic they forced the foreign company to breach their contract to supply enough medication to save peoples

### Economic Issues Simulation Paper

1485 words - 6 pages Health Care Reform Project, Part I Team HCS 440 Class University of Phoenix Online March 5, 2012 Health Care Reform Project, Part I Within the United States are many concerns about health care economics. Health care costs are continuing to rise and it contributes to the impact of providing the older population with medical care. “The aging of the U.S. population is translating into many more visits to

### Harvard Simulation Letter Essay

1870 words - 8 pages sunflower   utraceuticals n to Sunflower Nutraceuticals, Board of Directors from Teresita Alvarez, CEO re Working Capital Financing Options I wanted to update you on my efforts to secure an increased line of credit for working capital. Despite my repeated efforts and the calls that both of you have made to our bank’s senior officers, Miami Dade Merchant’s Bank (MDM) continues to be inflexible. It refuses to increase our \$3.2 million