Considering Are all linear programs convex?. Does that imply that whenever I can reformulate a real world problem into linear programming setting that I can get to the optimal solution with Dantzig's algorithm since the problem is convex? Or are there any further restrictions that need to be satisfied to get to the optimal solution?
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1Hi: the objective function needs to be convex, any constraints need to be linear and the set over which the function is being optimized needs to be convex also. – mark leeds Mar 16 '24 at 11:32
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1Yes, all linear programming problems over reals are convex, be it a minimization or a maximization problem. The obejctive is also linear, so it is both convex (for minimization purpose) and concave (for maximization purpose). – Marc Dinh Mar 20 '24 at 15:07
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If the objective function and constraints are all strictly convex, then you are guaranteed a globally (but not necessarily unique) optimal solution, and also consider this and this. Since linear programming problems are composed of only linear functions, and all linear functions are convex, then it would imply from the linked Q&As that any locally optimal solution you should find would be a globally optimal solution.
Thus, if you formed any real-world problem into a linear model and solved it, it would return a globally optimal solution. However, your mileage of the problem may vary as a model gives you insight into the problem but may not encompass everything depending on the assumptions you made to turn it into a linear model.
Miss Mae
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