## Main.DiscreteOptimization History

Changed line 40 from:
m.Obj(4*x1**2-4*x2*x1**2+x2**2+x1**2-x1+1)
to:
m.Minimize(4*x1**2-4*x2*x1**2+x2**2+x1**2-x1+1)
June 21, 2020, at 04:35 AM by 136.36.211.159 -
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The integer problem can be solved with [[https://gekko.readthedocs.io/en/latest/|Python GEKKO]]. The option '''integer'''=True is used to switch the variable from continuous to integer form. The APOPT solver is required to solve problem with integer variables but other solvers such as IPOPT can provide a relaxed solution where the integer conditions are not enforced.

(:source lang=python:)
from gekko import GEKKO
m = GEKKO() # create GEKKO model
# create integer variables
x1 = m.Var(integer=True,lb=-1,ub=1)
x2 = m.Var(integer=True,lb=-1,ub=2)
m.Obj(4*x1**2-4*x2*x1**2+x2**2+x1**2-x1+1)
m.options.SOLVER = 1 # APOPT solver
m.solve()
print('x1: ' + str(x1.value))
print('x2: ' + str(x2.value))
(:sourceend:)
March 15, 2016, at 03:32 PM by 45.56.3.173 -
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At first glance it might seem solving a discrete variable problem would be easier than a continuous problem. After all, for a variable within a given range, a set of discrete values within the range is finite whereas the number of continuous values is infinite. When searching for an optimum, it seems it would be easier to search from a finite set rather than from an infinite set.

This is not the case, however. Solving discrete problems is harder than continuous problems. This is because of the combinatorial explosion that occurs in all but the smallest problems. For example if we have two variables which can each take 10 values, we have 10*10 = 100 possibilities. If we have 10 variables that can each take 10 values, we have 10^10 possibilities. Even with the fastest computer, it would take a long time to evaluate all of these.
March 15, 2016, at 03:32 PM by 45.56.3.173 -
Changed lines 5-8 from:
to:
At first glance it might seem solving a discrete variable problem would be easier than a continuous problem. After all, for a variable within a given range, a set of discrete values within the range is finite whereas the number of continuous values is infinite. When searching for an optimum, it seems it would be easier to search from a finite set rather than from an infinite set.

This is not the case, however. Solving discrete problems is harder than continuous problems. This is because of the combinatorial explosion that occurs in all but the smallest problems. For example if we have two variables which can each take 10 values, we have 10*10 = 100 possibilities. If we have 10 variables that can each take 10 values, we have 10^10 possibilities. Even with the fastest computer, it would take a long time to evaluate all of these.

Discrete decision variables are those that have only certain levels or quantities that are acceptable at an optimal solution. Examples of discrete variables are binary (e.g. off/on or 0/1), integer (e.g. 4,5,6,7), or general discrete values that are not integer (e.g. 1/4 cm, 1/2 cm, 1 cm).
March 15, 2016, at 03:22 PM by 45.56.3.173 -

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December 19, 2014, at 02:22 PM by 107.188.175.164 -
** [[Attach:apopt_minlp.zip | Branch and Bound with APOPT solver (MATLAB and Python)]]
February 16, 2014, at 04:55 AM by 23.255.228.67 -
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One often encounters problems in which design variables must be selected from among a set of discrete values.  Examples of discrete variables include catalog or standard sizes (I beams, motors, springs, fasteners, pipes, etc.), materials, and variables which are naturally integers such as people, gear teeth, number of shells in a heat exchanger and number of distillation trays in a distillation column. Many engineering problems are discrete in nature.

Deleted lines 18-19:

One often encounters problems in which design variables must be selected from among a set of discrete values.  Examples of discrete variables include catalog or standard sizes (I beams, motors, springs, fasteners, pipes, etc.), materials, and variables which are naturally integers such as people, gear teeth, number of shells in a heat exchanger and number of distillation trays in a distillation column. Many engineering problems are discrete in nature.
February 15, 2014, at 03:29 PM by 23.255.228.67 -
Changed lines 11-12 from:
[[Attach:chap4_worksheet1.pdf | Attach:bnb_contour.png]]
to:

->
[[Attach:chap4_worksheet1.pdf | Attach:bnb_contour.png]]
February 15, 2014, at 03:26 PM by 23.255.228.67 -
Changed lines 9-11 from:
* [[Attach:chap4_worksheet1.pdf | Lecture 4.2: Branch and Bound Exercize]]
to:

* [[Attach:chap4_worksheet1.pdf | Lecture 4.2: Branch and Bound Exercise]]
[[Attach:chap4_worksheet1.pdf | Attach:bnb_contour.png
]]
February 20, 2013, at 09:50 PM by 128.187.97.21 -
* [[Attach:chap4_worksheet1.pdf | Lecture 4.2: Branch and Bound Exercize]]
February 19, 2013, at 09:48 PM by 128.187.97.21 -
Changed line 8 from:
** [[https://apmonitor.com/online/view_pass.php?f=minlp_apopt.apm | Branch and Bound]]  (Select APOPT solver)
to:
** [[https://apmonitor.com/online/view_pass.php?f=minlp_apopt.apm | Branch and Bound with APMonitor]]  (Select APOPT solver)
February 19, 2013, at 09:45 PM by 128.187.97.21 -
Changed line 8 from:
** [[https://apmonitor.com/online/view_pass.php?f=minlp_apopt.apm | Branch and Bound with APMonitor]]
to:
** [[https://apmonitor.com/online/view_pass.php?f=minlp_apopt.apm | Branch and Bound]]  (Select APOPT solver)
February 19, 2013, at 09:44 PM by 128.187.97.21 -
Changed lines 5-6 from:
* [[Attach:chap4_lecture_1.pdf | Chapter 4: Introduction Lecture]]
* [[Attach:chap4_discrete_opt.pdf | Chapter 4: Discrete Optimization Chapter]]
to:
* [[Attach:chap4_discrete_opt.pdf | Chapter 4: Discrete Optimization]]
* [[Attach:chap4_lecture_1.pdf | Lecture 4.1: Introduction to Discrete Optimization]]
** [[Attach:matlab_minlp.zip | Branch and Bound with MATLAB]]
** [[https://apmonitor.com/online/view_pass.php?f=minlp_apopt.apm | Branch and Bound with APMonitor
]]
February 19, 2013, at 09:38 PM by 128.187.97.21 -
Changed lines 5-7 from:
[[Attach:chap4_discrete_opt.pdf | Chapter 4: Discrete Optimization]]

[[Attach:chap4_lecture_1.pdf | Chapter 4: Lecture Notes - 1]]
to:
* [[Attach:chap4_lecture_1.pdf | Chapter 4: Introduction Lecture]]
* [[Attach:chap4_discrete_opt.pdf | Chapter 4: Discrete Optimization Chapter]]
February 19, 2013, at 09:36 PM by 128.187.97.21 -

[[Attach:chap4_lecture_1.pdf | Chapter 4: Lecture Notes - 1]]
January 11, 2013, at 02:23 PM by 69.169.188.188 -

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December 22, 2012, at 04:11 PM by 69.169.188.188 -