What algorithm can solve a linear minimization problem with given constraints?

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    Linear Minimization
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Discussion Overview

The discussion revolves around identifying the type of problem presented in a linear minimization context and exploring suitable algorithms for solving it. The focus is on linear programming, particularly in relation to constraints and the nature of the feasible region.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant inquires about the classification of the problem and potential algorithms for solution.
  • Another participant emphasizes the necessity of clearly defining what is being minimized to complete the question.
  • A third participant identifies the problem as a Linear Programming issue and suggests the Simplex Method as a straightforward solution.
  • A later reply discusses the properties of linear programming, noting that the minimum or maximum of a linear function occurs at a vertex of the feasible region, which is described as an n-dimensional rectangle based on the given constraints.
  • This reply also highlights potential complications with the constraints, particularly regarding the application of the condition S > 50 after determining values and the exclusion of certain faces from the feasible region.

Areas of Agreement / Disagreement

Participants generally agree that the problem is a linear programming issue, but there is no consensus on the completeness of the problem statement or the implications of the constraints.

Contextual Notes

There are limitations regarding the clarity of the minimization objective and the implications of the constraints on the feasible region, which remain unresolved.

specone
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Hi all,

does anyone know what type of problem this would be classified as, and what kind of algorithm I could use to get a solution?

minimize:
P=60w1+30w2+10w3+100w4

subject to:
P ≥ 50
0 < wi ≤ 1

thanks very much
 
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What are you trying to minimize? Without this, the question is incomplete. Defining what you want to minimize with these conditions is called linear programing.
 
It looks like a Linear Programming problem. The most straightforward solution algorithm is the Simplex Method.
 
The fundamental result in "linear programming" is that a minimum or maximum of a linear function, in a convex polyhedron, must occur at a vertex of the polhedron. Your condition, [itex]0< w_i\le 1[/itex], says that the "feasible region" is a n-dimensional rectangle and it is easy to calculate its [itex]2^n[/itex] vertices. However, the condition S> 50, on the target function will have to be applied after determining values. Also the fact that the faces [itex]0= w_i[/itex] are NOT in the feasible region can be a problem.
 

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