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'''Vector optimization''' is an [[optimization]] problem of simultaneously optimizing multiple [[objective function]]s subject to constraints and a given ordering. Any [[multi-objective optimization]] problem is a vector optimization problem with the trivial ordering.
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== Canonical example ==
In mathematical terms, the vector optimization problem can be written as:
:<math>C\operatorname{-}\min_{x \in M} f(x)</math>
where <math>f: X \to Z</math> for some [[vector space]]s <math>X,Z</math>, <math>M \subseteq X</math>, <math>C \subseteq Z</math> is an [[ordering cone]] in <math>Z</math>, and <math>C\operatorname{-}\min</math> denotes minimizing with respect to the ordering cone.
 
The solution to this minimization problem is the ''smallest'' set <math>S</math> such that for every <math>s \in S</math> there exists a <math>x \in M</math> where <math>f(x) = s</math> and <math>S + C \supseteq \{f(x): x \in M\}</math>.
 
== Solution types ==
* <math>\bar{x}</math> is a ''weakly efficient point'' (w-minimizer) if there exists a [[neighborhood (mathematics)|neighborhood]] <math>U</math> around <math>\bar{x}</math> such that for every <math>x \in U</math> it follows that <math>f(x) - f(\bar{x}) \not\in -\operatorname{int} C</math>.
* <math>\bar{x}</math> is an ''efficient point'' (e-minimizer) if there exists a neighborhood <math>U</math> around <math>\bar{x}</math> such that for every <math>x \in U</math> it follows that <math>f(x) - f(\bar{x}) \not\in -(C \backslash \{0\})</math>.
* <math>\bar{x}</math> is a ''properly efficient point'' (p-minimizer) if <math>\bar{x}</math> is a weakly efficient point with respect to a [[closure (mathematics)|closed]] [[convex cone|pointed convex cone]] <math>\tilde{C}</math> where <math>C \backslash \{0\} \subseteq \operatorname{int} \tilde{C}</math>.
 
Every p-minimizer is an e-minimizer.  And every e-minimizer is a w-minimizer.<ref name="scalar2vector">{{cite doi|10.1007/s10492-006-0002-1}}</ref>
 
== Solution methods ==
* [[Benson's algorithm]] for ''linear'' vector optimization problems<ref name="Lohne">{{cite book|title=Vector Optimization with Infimum and Supremum|author=Andreas Löhne|publisher=Springer|year=2011|isbn=9783642183508}}</ref>
 
== Relation to multi-objective optimization ==
Any multi-objective optimization problem can be written as
:<math>\mathbb{R}^d_+\operatorname{-}\min_{x \in M} f(x)</math>
where <math>f: X \to \mathbb{R}^d</math> and <math>\mathbb{R}^d_+</math> is the positive [[orthant]] of <math>\mathbb{R}^d</math>.  Thus the solution set of this vector optimization problem is given by the [[Pareto efficient]] points.
 
== References ==
{{Reflist}}
 
[[Category:Mathematical optimization]]

Revision as of 18:08, 22 February 2014

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First, of course, you need to get your income items together. Garage sales are an effective way to sell items that you will no longer have room for. Youd be astonished what doodads and miscellany your neighbors may wantmusty books, old gold use, framed pictures of long-lost third cousins, and other unmentionables that youve had no use for since who knows when. Much more shocking, your neighbors are willing to spend cold hard cash for these items. Which makes your garage sale a win-win proposition for you.

Come to think of it, however, a garage sale your own house could be a real job. It requires a Herculean effort to organize and collect every last little thing you want to offer. Then you definitely have to tag each item and keep track of it-in a hardcopy log. You must print up, build, and pay-for pamphlets, which have to be distributed o-n telephone poles and supermarket message boards over the region, to make sure that your neighbors show up with money in hand. The task can seem endless and your chances of creating a profit lean.

Dont misunderstand me. Your garage sale was a great idea, but while you can easily see, there are various restrictions to keeping one at your property the old-fashioned way. Get with the changing times. The garage sale has evolved because of the multitude of Web sites that enable you to sell your products online. People are selling more stuff than ever online because of the capability of Web classified web sites. To get different ways to look at it, we know you check out: account.

The Internet contains a few critical advantages over trying to sell your stuff the way. Whenever you evaluate these benefits to the difficulties with home-based garage income, youll see the difference:

ONLINE: Your products are available for purchase for an unlimited time, until they offer.

AT HOME: Your products are available until you call it quits for the night.

ONLINE: Rain, sleet, or snow, your income arent affected in the least.

AT HOME: Better have a rain day or a very large umbrella.

ONLINE: You should not constantly watch over-your goods or interact with uninterested consumers. Just wait until an individual contacts you by e-mail if they are ready to acquire.

AT HOME: Prepare yourself for a lot of time of staring across the street for customers to look. Be equipped for too many questions but too few purchases, when they do arrive.

ONLINE: Advertise free of charge to your pals and neighbors over email. To read additional information, please have a gander at: my garage doors oxnard.

AT HOME: Shell out money for pamphlets, after which you may get to know all the telephone poles in town.

ONLINE: Spend your breaks the direction they were designed to be: relaxing and enjoying your time with friends and family.

Your weekend is Spent by at HOME: hauling boxes, buying and distributing brochures, waking up early, and awaiting customers.

As you can see, its no even match. An online storage sale wins hands down.

Today that youre actually excited to get started with your garage sale, you should know that there"s one caveat. As easy as it might seem, your web garage sale will be even easier if you select the right classified site.

Your best bet is to do your re-search and find websites with the most features and benefits, including

(1) No sign-up, registration, or transaction fees.

(2) Free advertising record with text information and pictures.

(3) Long-term offer publishing and free renewals.

(4) Easy, accurate, electronic, and automatic record keeping.

The most effective classified websites have these incredible functions and then some. Not only will you be able to remove your domiciles clutter with the aid of these classified sites. Your garage sale will also park a pro-fit in your pocket..

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