For best results, training should be considered a full time job. One’s entire lifestyle must be directed towards this objective. This lifestyle will also require following the correct training program religiously over time.

All athletes are limited more by their minds more than their bodies

One needs to develop a winning attitude. One needs to combine physical fitness with mental fitness.

This chapter touches on a few mental opportunities for success:

• Motivation – Competitive cycling demands an inner drive to excel. Everything in your life must be focused on the goal
• Dreams, goals, and missions – high performance starts with a dream. This turns into a goal, and ultimately the mission. Are you willing to make any sacrifice, ride any number of miles, or do any workouts deemed beneficial to achieving the goal?
• Believe to achieve – there are bound to be setbacks. One needs to remain confident, patient and mentally tough to power through. Relive each day’s wins, big or small to boost self-confidence
• Mental toughness – when we’re at our limits, fatigue sets in, and the mind tries to make compromises. A fully committed rider gets through it
• Patience – success won’t come quickly. We can introduce incremental change, like: sleeping 30 minutes earlier each day, cutting out 10% of junk food from our diets, etc.
• Commitment and tenacity – there are moments where fatigue says to stop, but the fully commited and tenacious rider finds a way to keep going
• Consistency and routines – the mind and body like regular and standard schedules. Things like jobs, families, and other responsibilities may interfere
• Your high-performance team – surrounding each champion is a network of family, friends, teammates, coaches, etc.

This topic can get rather deep, so there is value in reading other books on the subject and speaking with other successful professionals from all walks of life.

By improving one’s mental approach to cycling, their training and racing will benefit greatly too.

Next: Chapter 2 – Physical performance
Chapter list: Chapter summaries

Personally, I simply got tired of sleeping so little.  I always sacrificed sleep for more work time in the past.  But now, I’ve come to realize how powerful being rested really is.  It feels so empowering when I wake up in the morning, fully refreshed.  I don’t have to struggle to get out of bed or set 5 alarm clocks to get me out of bed in the morning.  The reason that the struggle exists is because the body wants more sleep.  This is a clear sign of not sleeping enough.  I’m definitely no sleep expert and I don’t claim to be, but I thought I’d share some techniques in how to think differently about sleep.  But first, I think I should share my experience with sleep deprivation, which I felt heavily motivated me into adopting this way of life.  All of this below is just my view on sleep health.

My Battle with Sleep Deprivation

In high school, I pretty much got 8 hours of sleep a night.  It wasn’t that hard since at that time I didn’t really think about optimizing my sleep.  I just remember being nagged to go to sleep by my parents every night.  I never had to stay up doing any homework or anything like that.  Even my first year of university was like that.  Rarely found myself staying up to do work unless there was a exam the next day or something.

Second year of university is when it all changed.  A thought clicked in my head that I could sleep less and get more work done.  It’s not like I felt sleepy at night anyways.  For a while it was alright, but then as I slept less and less, my sleep debt accumulated.  I found myself napping in class.  I even micro-slept while I drove and almost got into not one, but a couple car accidents.  I didn’t try to do anything about it at the time.  I liked the extra time.  There was a while where I slept about 4.5 hours a night because I stayed up playing computer games.  I looked like a zombie.  I yawned all the time and everyone knew I was sleepy.

In my third year, I lived alone on campus.  No one told me when to wake up to go to class.  With all the sleep debt I had, I struggled to get up every morning.  Eventually, I stopped going to morning classes altogether and my grades suffered.  I started questioning myself, “why is it so hard to sleep early?”  I was easily annoyed all the time.  I was so tired at one point, that I caught myself hallucinating.  It was weird because I saw this strange bug crawling up the curtain and when I waved the curtain around, it was gone.  I couldn’t find it anywhere on the floor.  I wanted to change my sleeping habits, but I couldn’t do it.  Logic told me to do it, but I couldn’t act on it.

My sleeping habits didn’t change until I went to Germany for my co-op work term.  I guess because I was adjusting to this new environment, I actually felt sleepy at night.  This urge made me sleep a lot earlier than what I was used to doing.  I felt rested for the first time in a long time.  It felt so good.  My constant zombie look faded away.  I could tell that my skin complexion was improving.

I guess because I’ve seen how dangerous sleep deprivation could be, it finally compelled me to change my ways.  Ever since I’ve made sleep a priority, I feel a lot healthier.  I lose out on wake-time for working, but I am able to think better when I am rested.  My thoughts are a lot clearer.  That makes me a more efficient worker.

Getting into the Frame of Mind

You’ve got to be truly convinced that you want to get more sleep.  I know a lot of people who say things “yeah I know I don’t get enough sleep”, but they don’t do anything about it.  Nothing at all.  Everyday they keep their usual sleep routine and they’re struggling all the time.  They’re the ones who are always late to morning appointments, and some even for afternoon appointments.  Why does it have to be that way?  It doesn’t.  But before even trying to maintain a proper sleep routine, people really have to change the way they think about sleep.

Sleep is NOT a waste of time. I’ve heard people say, “sleep is a waste of time” or “sleep is for the weak”.  It definitely is not.  Somehow I think there is a competition between who gets the least amount of sleep and still gets the most work done.  I’ve heard people people saying (sometimes bragging) stuff like how they got only 3 hours of sleep and then pulled an all-nighter to study for an exam.  Sure, they definitely were up for a while, but should we question the efficiency of that work?  Of course.  Anybody who has pulled an all-nighter has experiences how “out of it” they were by the end if not half way through.  Nobody wants to do an all-nighter.  That is an extreme case though.  A more common theme is to stay up for 18 hours or so.  That leaves 6 hours to sleep.  The average for most people lies somewhere around 7.5-8 hours.  According to the statistics, more than half of us need more than 6.  By not getting our required amount of sleep, this so-called “sleep debt” will accumulate over time.  Like I mentioned before, I once slept on average 4.5 to 5 hours a night for about 3 months straight.  People around me could tell I looked like a zombie.  I’ve definitely experienced sleep debt first hand.  In retrospect, doing this was so stupid, but that experience definitely helped me reshape what I thought about sleep.  It is a necessity.

Getting into the right frame of mind takes time.  I definitely did not just one day choose to make sleep a priority.  It happened over time.  For me it took a few years in order for me to drill it into my head.  After sleeping late and waking up early, I’d ask myself why I did that.  Why did I sleep so little the night before when the consequence was me feeling sleep deprived the day after?  As I’m in my early 20’s, sleeping late is easy.  It’s not difficult to just stay up late doing nothing in particular.  In fact, it seems to be natural tendency for most people in my age group to be night owls.

Going to bed early is not a chore.  Chores are something we’d rather not do, but we’re only doing it because we have to.  This attitude will not work.  Eventually, you’ll give up and revert to your old ways.  You have to change your attitude toward sleep.  You know how some people feel passionate about their work and will always feel motivated to do it?  That’s their motivation.  There is a similar motivation for sleeping early.  I want to go to sleep early today, so I feel rested and ready to go tomorrow.

Ask yourself why your sleep habits are the way they are right now.  Why do I have to wake up feeling terrible every morning?  Why do I have to set so many alarms just to get out of bed?  Why am I pressing the snooze button so much?  Shouldn’t I be getting up super early? Doesn’t this struggle to wake up give me less time to get ready?  If I don’t have enough time to get ready, won’t I look like a mess when I go to work/school?  Am I sleeping late all the time because I have so much work to do?  Could I be improving on my time management to make me more efficient? Is sleeping late worth it?  I’m sure there are way more questions to be asked, but asking yourself these questions will tell you how you really feel about it.  Everyone’s going to have different answers.  Some might want to change, but others might feel that what they’re doing is the best solution right now.

The Bottom Line

Sleep is absolutely essential for good health and its benefits are far reaching.  Ever since I’ve slept better, I’ve been able to accomplish more.  No longer am I the sleep deprived zombie.  My grades in school are better despite harder courses and more commitments. Better sleep health is something that anybody can do, but they have to want it.  You have to convince yourself that sleep deprivation is a terrible thing and it is something to avoid.  You’ve got to get into the frame of mind that sleep health is good and the will to change.  If you truly want to improve your sleep habits for the better, the improvements will happen.

• Introduction
• Motivation
• Newton’s Method
• Steepest Descent Method
• Limitation
• Sources

Introduction

In many sciences, the problem of non-linear optimization appears quite often.  What happens if they’re non-linear?  Sometimes, the system of equations remains easy to solve even given non-linearity, but other times, the possibility of solving the problem analytically may not exist if the algebra gets too complicated.  The solution lies in using numerical methods (approximations).  Here, I present two common methods: Newton’s Method, and the Steepest Descent Method. 

Each method has its own limitations and the choice of which method to use depends on the problem.  These methods involve using iterative computation in order to reach a solution.  Luckily, computers can prove useful in providing solutions very quickly due to its quick processing power for numerical computation.

In order to understand some of the concepts and terms used in this article, the reader has knowledge in multivariable calculus.  Also, equations in picture form made the line spacing look funny, so I’m going to do my best write the equations as text.  I should also point out that I couldn’t write the vector arrow on top of the variables in text form so vectors are written in bold.

Motivation

In calculus, the idea of optimization involves taking the derivative of the function and equating it to zero.  In multivariable calculus, the same idea applies except, we set all the partial derivatives to zero and solve the system.

So in general, if we are given a function:

ƒ(x), where x = [x₁, x₂, … x_n]^T

The maximum or minimum can be found by taking a partial derivative with respect to the vector x.  This gives the following equations:

∂ƒ/dx₁ = 0

∂ƒ/dx₂ = 0

…

∂ƒ/dx_n = 0

A short-hand way of expressing the above equations is by using the gradient vector, giving∇ƒ(x) = 0.  By definition, the gradient is defined by:

∇ƒ(x) = [∂ƒ/dx₁, ∂ƒ/dx₂, … , ∂ƒ/dx_n].

An analytical solution is not always possible, hence the motivation for approximating the solution using numerical methods.

Newton’s Method

In general, this method is written below.  Explanation of each step will follow.

1. Given the equations to solve, derive equations for the approximating line or plane, depending on whether the problem is 2D, 3D, nD, etc.
2. Using the linearized equations, solve the system of equations
3. Using the results from step 2, generate new equations for a new approximating line or plane, and repeat step 2.

This method also involves starting with a guess, denoted by a variable with a subscript of 0.  A guess is required, because this is the starting point for this method.  Using this guess, each successive solution should converge (more on that later).

Step 1

In order to solve the equations ∇ƒ(x) = 0 (see Motivation for more information), we can apply multivariate Newton’s method to approximate the solution.  Like the single-variate (two-dimensional) version, a linear approximation is required.  However, instead of having an approximation in the form of a line, the approximation is a plane instead.

2D (line): ƒ(x) ≈ ƒ(x₀) + ƒ’(x₀)(x – x₀)

3D (plane): ƒ(x, y) ≈ ƒ(x₀, y₀) + ƒ_x(x₀, y₀)(x – x₀) + ƒ_y(x₀, y₀)(y – y₀)

nD (nD plane): ƒ(x) ≈ ƒ(x₀) + ∇ƒ(x₀)(x – x₀)

Since we are trying to optimize, we have the partial derivatives of the function we are trying to optimize, but in order to apply Newton’s Method, we need a linear approximation of the partial derivatives.  This means that in order to approximate the partial derivatives, we need to take another derivative in order to get the slopes of the approximating planes.  If that description didn’t make sense, try the following.  For an equation with two independent variables, there will be two equations for the partial derivatives, one for x and one for y.  Taking partial derivatives from each of those equations again will give the slopes of the plane that approximate the first partial derivative.  So if we have two independent variables, we end up getting four equations: ƒ_xx, ƒ_xy,ƒ_yy, and ƒ_yx (ƒ_xy and ƒ_yx should be the same).  In general, for n partial derivatives, we would have n² equations if we took another set of derivatives.  The approximations for both of the partial derivatives would be:

ƒ_x(x, y) ≈ ƒ_x(x₀, y₀) + ƒ_xx(x₀, y₀)(x – x₀) + ƒ_xy(x₀, y₀)(y – y₀)

ƒ_y(x, y) ≈ ƒ_y(x₀, y₀) + ƒ_yy(x₀, y₀)(x – x₀) + ƒ_yx(x₀, y₀)(y – y₀)

Plug in the values for the second derivatives, and the values from the initial guess (the x y values and what function value you get using those values).

Step 2

Although the equations generated from step 1 is yet another system of equations, the main difference is that the equations from step 1 are linear.  This means that the system can be solved analytically.  Solve for the solution of the system using row reduction methods, or other methods.  The solution from solving the system will be the values used for the next iteration and will replace the guessed values.  The solution may or may not be close to the actual solution.  More iterations need to be done to see if the answer changes with each iteration.  Usually if the guess is really far away, then it takes a lot of iterations to get it really close.  However, sometimes, a bad guess may cause the solution to diverge, meaning that the solution will be further and further off.

Step 3

Generate new equations for the approximating planes using the solution from step 2.  Using these equations, go back to step 2 and solve it again.  This method is super tedious if done on paper.  Matlab can help, but then again, Matlab can do the entire approximation with just a few commands, but not everyone has access to it unfortunately.

Hopefully after a few iterations, the solution stops changing.  When that happens, you can substitute the final solution back into the original equation to see if it works!

Steepest Descent Method

Here is the general method:

1. Find the direction of the maximum or minimum directional derivative
2. Follow it until a maximum or minimum is reached

Newton’s Method is sufficient in a lot of cases, but sometimes it may be too hard to go beyond first derivatives.  This main idea of this method is to use the first derivative and follow the direction in which the function gets larger/smaller.  A fundamental concept used in this method is the directional derivative.  Since in 3D (or higher dimensions), we are not restricted to movement along the x axis.  This means we could move in any random direction, and in each direction the rate of change is going to be different.  To calculate the max or min, we need to calculate where the directional derivative is largest or smallest.  Just like a boulder rolling down the steepest slope on the hill.

Starting with a guess, we can find out the value of the function at the point and the direction to travel in.  Take a step in that direction and see how the value of the function changed.  Keep doing that until a max or min is reached.

Step 1

The definition of the directional derivative is the dot product of the gradient and the unit vector specifying the direction.  Here is the definition of the directional derivative:

D_uƒ = ∇ƒ(x)·u

Since the definition involves a dot product, it is true that if the two vectors are orthogonal, the resulting dot product is zero.  The direction that this derivative is in is called the level direction.  But if the two vectors are parallel, we get the highest value of the directional derivative.  For a vector that is parallel in the same direction, the directional derivative will be a maximum, but if the vector is antiparallel (parallel but both pointing opposite ways), the directional derivative will be a minimum.

Since the maximum value of the directional derivative is parallel to the gradient, we can calculate the unit vector of the gradient and use it as the unit vector u.  In order to get the minimum, simply flip the direction of the vector by multiplying it by –1.

u = ∇ƒ(x)/|∇ƒ(x)|

Although not necessarily required, substituting this value into the definition of the directional derivative, and doing some simplification with algebra, the maximum and minimum directional derivative is given by:

Max D_uƒ = |∇ƒ(x)|

Min D_uƒ = -|∇ƒ(x)|

Using the initial guess, calculate the direction of to travel in at that point.

Step 2

Knowing the direction to travel choose a desired step size “h”.  This means that if the equation will be traversed in a particular direction, we need to know exactly how far to go.  Choosing smaller step sizes gives better computational accuracy but increases computation time.

To get to the maximum, use the direction for the maximum directional derivative.  To get the minimum, use the direction for the smallest directional derivative.

Starting with a guess, calculate the function value at that point.  Then take a step in the direction given by Step 1 using the step size.  The variable “u” here is a component of the unit vector

x_n+1 = x_n + u_xh

y_n+1 = y_n + u_yh

At this new value, calculate the function value again.  Keep repeating this until the maximum or minimum is reached.

Limitation

The approximation is generally only as good as the initial guess.  Sometimes, a bad guess can lead to a solution that never converges.  In that case, a different value must be used in order to find a converging solution.  For example, if we have a fourth order polynomial function, a local extreme will exist.  As x approaches –∞ or ∞, y will do the same.  If I use the steepest descent and give a bad guess, my solution will go to ∞ instead of the local extreme.

Sources

1. Calculus by Gilbert Strang
2. Applied Regression Analysis by Norman Draper and Harry Smith