Unbounded and Bounded Linear Program optimal solutions..?
The answers are: a) True; b) False. a) Assume the opposite - let the feasible region is bounded. If it is not closed, add all the boundary points to it. Now the objective function, being a continuous linear function, defined in a closed and bounded domain, surely attains its maximum and minimum values according Bolzano-Weierstrass Theorem (follow the link in Sources below for details). Hence the optimal value can not be infinite - contradiction. If the feasible region is bounded, but not closed, all values are still bounded, but the optimum may not be attained. b) Consider {z = x + y}, subject to 1 ≤ x + y ≤ 2. The feasible region is the stripe between the lines x + y = 1 and x + y = 2 (unbounded region, I recommend to graph it). Now z_min = 1, attained in each point on the 1st line, z_max = 2, attained on the 2nd, so both optimal solutions are bounded.
How do I calculate extreme points for an optimal solution?
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This has to do with the Fundamental theorem of linear programming.In mathematical optimization, the fundamental theorem of linear programming states, in a weak formulation, that the maxima and minima of a linear function over a convex polygonal region occur at the region's corners. Further, if an extreme value occurs at two corners, then it must also occur everywhere on the line segment between them. (From Wikipedia)See this for best possible explanation, Wolfram Demonstrations Project
This question is a little unclear. When does an optimization problem not have an optimal solution? Only when the function value is not defined (i.e. infinity or -infinity) at some point within its domain (domain = valid input values) , or when the function does have an optimal solution but it is achieved "at infinity", so no point within its domain achieves that value. Suppose the function has a finite closed domain and well defined at all points. Then, any optimization problem definitely has an optimal solution and it is achieved within the domain (because the domain is closed). I'll assume you meant "When can you efficiently find the optimal solution?"Minimization problems concerning a convex function have an optimal solution that can be found efficiently. A convex function is one which is bowl-shaped, then the minimum is achieved at the bottom of the bowl, and a procedure like gradient descent can help you find it. I won't discuss the proof here because you can find it in many places, like here: http://www.seas.ucla.edu/~vanden... Another question you may have meant is "When is this optimal solution unique?" That happens, for example, for strictly convex functions defined on a convex set. See If $f$ is strictly convex in a convex set, show it has no more than 1 minimum
Please help with finding the feasible region and extreme points of the feasible region?
the 1st answer is right, yet whilst the question calls so you might supply the two police vehicles and firetrucks, then 50 police vehicles and 40 firetrucks isn't maximum appropriate because of the fact they don't extra healthful the regulations. enable p = # of police vehicles, f = # of hearth tucks all of us be attentive to p + f = or below ninety all of us be attentive to 2p + 7f = or below 350 or the generating time must be below or equivalent to 370 hrs. all of us be attentive to 2p + 5f = or below 270 or the generating time must be below or equivalent to 370 hrs. so if we subtract (2p + 7f = 350) - (2p + 5f = 270) = (2f = 80) then f= 40 Then positioned f decrease back in between the above eqs. 2p + 5(40) = 270 so p=35. 35 + 40 < ninety, so it meets the regulations.
It depends on whether it’s right and what your goals are.Ultimately using the optimal solution is best, but it may not be realistic to implement due to resistance against it, especially if it is considered an extreme plan. In such a solution, presenting a plan that is an inprovement, but perhaps more palatable to those who oppose the more extreme plan.The middle road will not be the best simply because it is the middle road—the idea is fallacious. If the “Right” in your country is Fascist and the “Left” is Oligarchical then a solution to greater political rights for those that are middle and lower class will not be in the middle of the two position.Those who identify as Moderates simply because they believe Moderate equals good 100% of the time are naive, and it is best to choose what is best based off of evidence, and if that evidence leads you to what is considered extreme then perhaps it is simply the established political norms that are incorrect.
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Why is it that "the optimal allocation of the activity occurs where Marginal beneft=marginal cost?"?
Well, would you agree that you want the maximum benefit? That is the obvious goal, right? However, we cannot achieve any benefits at all without some cost. Total utility = Total Benefits - Total Cost In calculus, if we want to maximize something, we find where the derivative is equal to zero. Then it is a maximum: This gives: Marginal Benefit - Marginal Cost = 0 or Marginal Benefit = Marginal Cost Now this only works if you can take derivatives of both functions and that you can take yet another derivative and find if is a maximum or minimum. This means that both functions have to be 'twice differentiable'. But who cares about the math, why is the marginal conditions work? It works because it assumes that marginal costs is always rising and the marginal benefit falling, so that there is a unique solution. That solution is where the extra cost just equals the extra benefit. This makes sense because if you got another quantity, it costs more than it gives you benefits, so you shouldn't do it. Similarly, if the marginal cost is less than the benefit, then it makes sense to continue. In short, the marginal decision tells you the direction of the total. Again, this is only true if we assume the proper conditions are met to make it so.
For Linear programming, how would you find the optimal solution or?
The feasible region is in a plane, so it's easy to give an answer in this case. The theory of linear optimisation says that the maximum (and the minimum) value of the objective function p will be found on the boundary of the feasible region. More specifically, it will occur at one of the vertices of the polygon formed by the feasible region (and possibly on one of the sides, if the max occurs at two adjacent vertices). So here is an easy way to find the maximum value of p. It helps if you draw the feasible region, to help keep track of what you are doing. Determine first the points of intersection of the constraints (basically, the vertices of the polygon formed), by solving the corresponding equations. You will then have a set of points (x,y) representing the boundary of the feasible region. Then, just plug each of these points into the objective function p, and see which one gives the maximum value. If two vertices give the same value, then every point on the side between them will also give that value.
Put succinctly (at least by my standards), the simplex method starts with a feasible but suboptimal solution and generates a sequence of feasible but less suboptimal ones until it reaches an optimal solution and stops. The dual simplex method starts with a superoptimal (too good to be true) but infeasible solution and generates a sequence of progressively less infeasible (and less superoptimal) ones until it arrives at a feasible solution (which will be optimal).For some LPs primal simplex is faster, while for some other LPs dual simplex is faster. Dual simplex is popular in situations where you start with an optimal solution and then add a few constraints (or modify the right hand sides of a few existing constraints), rendering the previous solution infeasible but superoptimal. That describes the branching process in branch-and-bound/branch-and-cut algorithms for mixed integer linear programs.