View source for Talk:Mathematical optimization

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==Table of Contents missing==

[[User:BumbleMath]] wrote: Why is there no ToC in this article? It should be generated automatically due to its length according to Wikipedia documentation.  <!--Template:Undated--><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|undated]] comment added 18:54, 2 December 2023 (UTC)</small> <!--Autosigned by SineBot-->
:Hi {{Ping|BumbleMath}} If you are using Wikipedia at the Desktop, the TOC has moved to the upper left corner (three bars) and can be extended. At [[Special:GlobalPreferences]] you can switch to another skin. Greetings [[User:Bigbossfarin|Bigbossfarin]] ([[User talk:Bigbossfarin|talk]]) 02:16, 4 December 2023 (UTC)

:Hi @[[User:Bigbossfarin|Bigbossfarin]], oh, thanks. I have to admit it is there (but very hidden though). Thanks for your reply. [[User:BumbleMath|BumbleMath]] ([[User talk:BumbleMath|talk]]) 06:09, 4 December 2023 (UTC)

== Argmin ==

Sorry to serious math people for any errors in my explanation of min, arg min, etc..

Perhaps one of you could also clarify for me what the subscript x in ''min<sub>x</sub> f(x)'' means. (I take this to mean, hopefully correctly, "What is the minimum value of f(x)?") Is it an indication that x is a variable, so you don't accidentally treat it as a constant? If so, when do you have to specify subscript x and when do you not?

--[[User:Ryguasu|Ryguasu]] 04:22 Nov 6, 2002 (UTC)

It's in the context of argmin, argmax, then it's the value of x which minimizes f(X)

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The last example of section 1 is not correct. [[User:Patrick|Patrick]] 02:07 Dec 20, 2002 (UTC)


==Combinatorial optimization==

[[User:Artur adib]] wrote: 

:In  [[mathematics]], the term '''optimization''' typically refers to the study of ...

I would say not typically, but all the time. So the word ''typically'' is reduntant.

: (When the set ''A'' contains a finite number of elements, the problem falls in the domain of [[combinatorial optimization]].)

OK, you do have a point. But you see, to include all the special cases would make the definition way too long. For example, I specialize in infinite dimensional optimization, when the domain of the objective function is a space of functions, or a space of shapes. But you see, if we add your special case, and my special case, what becomes is a mess. So, for the sake of clarity, and at the expence of being the most general, I deleted your sentence. There is still that link at the bottom about combinatorial optimization you put. You can also add it in the
''Major subfields'' section, several paragraphs below the definition of optimization. But I would suggest that we keep the definition of optimization simple. I would be very interested in hearing what you think about this. Thanks! --[[User:Oleg Alexandrov|Oleg Alexandrov]] 23:01, 18 Dec 2004 (UTC)


== General formulation and notations==

Optimization problems in finite dimension can assume the following general form

:<math> \min\limits_{x,y}\;\; Z = f(x,y) </math>

:<math> \mbox{subject to: }\ </math>

::<math> h(x,y) = a\ </math>
 
::<math> g(x,y) \leq b </math>

::<math> x^{L} \leq x \leq x^{U}, x \in \Re^{n} </math>

::<math> y \in \{0,1\}^{p} </math>

Where ''f'' is a scalar function of the continuous variables ''x'' and of the discrete variables ''y'', ''h'' is a ''q''-dimension vector function representing equality constraints, ''g'' is a ''m''-dimension vector function representing inequality constraints and ''x''<sup>''L''</sup> and ''x''<sup>''U''</sup> are lower and upper bounds on the ''x'' variable, respectively. Generally, the inequality constraints do not necessarily need to formulated with an upper limit to their value (≤), but this formulation constitutes a general formalism once any constraints in the form of a lower bound (≥) can be easily converted to an upper bound. Rigorously, a problem containing only the inequality constraints would, by itself, be a general representation of a mathematical programming problem, once the bounds on the variables can be immediately represented by this form and equality constraints can be represented by the association of two simultaneous constraints having the same upper and lower bounds.

Optimization problems are divided into distinct classes, according to characteristics of the functions and variables that compose them. In the case in which all the functions that belong to the model are [[linear function]]s and the set of discrete variables is empty (''q = 0''), the problem belongs to the class of [[linear programming|Linear Programming]] (LP) problems. If, conversely, any of the functions in the problem present non-linearity and it’s variables are still all in a continuous space, the problem belongs to the class of [[Nonlinear programming|Non-Linear Programming]] (NLP) problems. The special case in which the objective function of a continuous problem contains quadratic and bilinear terms and the entire set of constraints is composed of linear functions is regarded as [[Quadratic programming|Quadratic Programming]] (QP). Finally, if the problem contains discrete variables (''q > 0''), it belongs to the classes of [[Mixed Integer Linear Problems]] (MILP), if it is composed of linear functions, and [[Mixed Integer Non-Linear Problems]] (MINLP), otherwise.

== Calculus of Variations ==

Calculus of variations is about sending an action integral over some space to an extremum by varying a function of the coordinates. It may be used in the context of temporal trajectories but that is not the definition of it and is not even the definition given by the Wikipedia page. The suggestion that Calculus of Variations is solely used to compute trajectories through time is misleading.

{{Resolved}}