(optimization, computer science:)
The replacement of discrete constraints with continuous ones in a problem (and solving that instead).

Integer constraint problems (optimising when one or more variables are constrained to be integers) are notoriously difficult: integer programming (which is linear programming with integer variables) is NP hard, whereas we know how to solve linear programming in polynomial time. So it can make sense simply to remove all integer constraints from variables, and seek an optimal solution with fractional values. Rounding appropriately, the result can be considered an approximation to the integer problem.

Ten steps to deep relaxation

Our lives are full of stress and pressures, some we create ourselves, others we cannot control. Coping with the stresses of modern life is an essential skill and there are many techniques available to us. This is one method that I found that works for me.

With your eyes closed, imagine you are standing at the top of a flight of ten steps. These are very safe steps. We're going to go down these steps together. At the bottom, you are going to be completely relaxed.

On the first step down, I want you to notice your breathing. Keep it as even and as steady as you can and remember that on the "out" breath you can increase the level of relaxation a great deal.

On the second step down, keep your breath even. On the "out" breath, relax your legs and your arms completely so they hang like heavy weights, floppy, heavy limbs, almost disassociated from you. Breathe away the tension on every "out" breath, even and steady, heavy and relaxed.

On the third step down, I want you to think about your buttocks, stomach, and chest. See if you can make them even more relaxed. On each "out" breath, let go of the stomach and midriff so they are completely relaxed and quiet breathing even and steady.

On the fourth step down, I want you to concentrate on the muscles in your face and jaw. On the next "out" breath loosen your jaw. Let your jaw hang open and breathe away the tension in your jaw, your temples and all the muscles around your eyes. Make your forehead smooth, keeping your breathing even and steady.

On the fifth step down, I want you to relax, from the top of your head, right down to your spine. Imagine the tension flowing like water over the top of your head, down the back of your neck and down to your spine. Let yourself rest comfortably in a chair, allowing the chair to support all of your weight, keeping your breathing even and steady.

On the sixth step down, I want you to unwind any differences between the two sides of your body. See an imaginary line running down between your nose, right down the center of your body, and see if the two sides are perfectly balanced.  If you can detect that one part is more tense than the other, on the next "out" breath, concentrate on the body part, reducing it further and further, letting the tension relax away.  Make it the same as the good side, relaxing further and further.

On the seventh step down, I want you to start relaxing your mind.  Focus your attention on a very soothing, peaceful image, far away from here.  If other thoughts come to your mind, gently push them off to one side.  You can deal with them some other time.  Absorb yourself totally into this one single soothing image.  Breathing "out" evenly and steadily, relaxing further and further, holding that image in your mind's eye and exploring it.

On the eighth step down, I want you to say to yourself on each "out" breath, some word you associate with deep relaxation, such as you are feeling at this moment.  It might be "relax", "soothe", "letting go", or it might be a word that reminds you of an image that is most soothing for you.  It should be a word that can evoke for you this beautiful, peaceful, balanced state that you can achieve for yourself.

On the ninth step down, return to your peaceful image, hold it in your mind, and just enjoy this feeling that you now have.  Your body is now completely relaxed and balanced.  There is nothing you need to think about except your soothing scene.

Finally, you step down to the bottom of the stairs, completely relaxed and peaceful.  Your body is heavy and tired.  Your mind is well-balanced, holding that peaceful image.  When you turn to come back up these stairs, you will find that you have retained this feeling, that you will go on feeling peaceful and alert all day, feeling as though you have just had a brief nap

As you walk back up these steps, gradually arouse your body as you go.

Nine...move your feet and toes.

Eight...move your fingers a little.

Seven...stretch out your arms and feel the stretch.

Six...move your legs.  Stretch your legs a little bit.

Five...move your shoulders. 

Four...move your head a little bit.

Three...take a deep breath and exhale.

Two...open your eyes.

One...come back to where you began at the top of the stairs.

Learning Relaxation Skills

My husband and I have attended many relaxation seminars and classes over the years and have obtained many relaxation skills.  The following is a guide that we have adapted to fit our needs for teaching people how to learn relaxation skills.

  1. Learn how to become deeply relaxed.  This applies to the muscles directly involved with your symptomatic areas as well as the muscles throughout all of your body.  In some muscles, such as the hands and arms, you can feel small differences in the muscle's tension.  In other muscles such as those found in your neck, head, and back, you are less able to feel small differences in muscle tension.  How deeply you need to relax varies from person to person.  You cannot predict how deeply you need to relax to prevent, reduce, or stop your symptoms.  It is generally better to reach deeper levels of relaxation.

  2. Learn to become deeply relaxed rapidly.  This will lower your tension level much quicker.  To stress the importance of this, think of yourself carrying a small suitcase that is very heavy.  You begin to sense that it would be better to relax your arms.  It makes sense to put everything down at once rather than put it down little bits at a time.  Muscles in the head, face, and neck need relaxation the same as any other muscles.  This is harder to do with the neck and head muscles.  Many of the muscles in the head and face are thinner, smaller, and not as strong.  They are used frequently but cannot tolerate as much tension as other muscles throughout the body.  Your goal should be to become more aware of the tension in these muscles and to be able to release the tension quickly.  This will allow the other muscles in your body to relax as well.

  3. Relax for long periods.  This goal depends on how rapidly and deeply relaxed you have been getting.  So, if you are lowering your bodies tension to slow and not relaxing deep enough, then you will likely have to spend more time to achieve the same results.  As your relaxation skills improve it will take less time to get into a deeper relaxation. 

  4. Relax frequently.  When you are carrying a heavy load, it would not make sense to put it down only a few times and carry it for longer periods.  In other words, relaxation should be done more frequently instead of carrying around tension for long amount of periods.  More of a balance between tension and relaxation is needed.  Many periods with a variety of length are vital to help you become aware of tension and relaxation. 

  5. Relax at the needed times for the appropriate situations.  Frequent relaxation periods are not enough.  You have to use your relaxation skills when and where you need them.  Use your relaxation skills before, during, and immediately after situations of increased tension.  You should use relaxation in all kinds of situations, even if you do not feel tense or aroused.  Many times tension or stimulation of the nervous system occur without your awareness.

  6. Increase your awareness of undo and unneeded tension.  As you become more aware of tension you will gain better skills to help avoid excess and unneeded tension.  Avoid excess, prolonged, and unnecessary tension.

  7. Increase your confidence in your abilities to relax, and your success at accomplishing the before mentioned steps.  Practicing, both alone and with others around, helps to prepare you for the situations when your relaxations skills will be most helpful.  Confidence in your skills, not only enhances the relaxation, but can actually help to prevent some tension.  As you experience relaxation effects when you are alone or with others you will gain confidence in yourself.  You will have developed ways to control your body in ways that you didn't think you could before.

Seven Steps to Relaxation

  1. You will be able to recognize tension in each large voluntary muscle.

  2. You will be able to recognize the sensation of relaxation in each large voluntary muscle.

  3. You will be able to control your relaxation by "letting go" of that muscles tension.

  4. You will experience deep muscle relaxation.

  5. You will be able to recognize sensations of tension and relaxation in the tiny eye and throat muscles.

  6. You will be able to control both eye and throat relaxation by "letting go" of the tension.

  7. You will be able to relax your whole body without having to first tense each individual muscle.

Source:  Terry, M.D., John. "Learning the skills and steps to relaxation.." . Springhaven Medical Center, Bluefield. 2001.

Re`lax*a"tion [L. relaxatio; cf. F. relaxation.]


The act or process of relaxing, or the state of being relaxed; as, relaxation of the muscles; relaxation of a law.


Remission from attention and effort; indulgence in recreation, diversion, or amusement.

"Hours of careless relaxation."



© Webster 1913.

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