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  2. LP-type problem - Wikipedia

    en.wikipedia.org/wiki/LP-type_problem

    In this formulation, the set S is the set of all vertices in both polytopes, and the function value f(A) is the negation of the smallest distance between the convex hulls of the two subsets A of vertices in the two polytopes. The combinatorial dimension of the problem is d + 1 if the two polytopes are disjoint, or d + 2 if they have a nonempty ...

  3. Linear programming - Wikipedia

    en.wikipedia.org/wiki/Linear_programming

    Let S 1 be the selling price of wheat and S 2 be the selling price of barley, per hectare. If we denote the area of land planted with wheat and barley by x 1 and x 2 respectively, then profit can be maximized by choosing optimal values for x 1 and x 2. This problem can be expressed with the following linear programming problem in the standard form:

  4. Linear-fractional programming - Wikipedia

    en.wikipedia.org/wiki/Linear-fractional_programming

    A linear program can be regarded as a special case of a linear-fractional program in which the denominator is the constant function 1. Formally, a linear-fractional program is defined as the problem of maximizing (or minimizing) a ratio of affine functions over a polyhedron ,

  5. Dual linear program - Wikipedia

    en.wikipedia.org/wiki/Dual_linear_program

    We use this example to illustrate the proof of the weak duality theorem. Suppose that, in the primal LP, we want to get an upper bound on the objective 3 x 1 + 4 x 2 {\displaystyle 3x_{1}+4x_{2}} . We can use the constraint multiplied by some coefficient, say y 1 {\displaystyle y_{1}} .

  6. Linear programming relaxation - Wikipedia

    en.wikipedia.org/wiki/Linear_programming_relaxation

    However, there is a fractional solution in which each set is assigned the weight 1/2, and for which the total value of the objective function is 3/2. Thus, in this example, the linear programming relaxation has a value differing from that of the unrelaxed 0–1 integer program.

  7. Basic feasible solution - Wikipedia

    en.wikipedia.org/wiki/Basic_feasible_solution

    For example, if is non-basic and its coefficient in is positive, then increasing it above 0 may make larger. If it is possible to do so without violating other constraints, then the increased variable becomes basic (it "enters the basis"), while some basic variable is decreased to 0 to keep the equality constraints and thus becomes non-basic ...

  8. Revised simplex method - Wikipedia

    en.wikipedia.org/wiki/Revised_simplex_method

    x B must be correspondingly decreased by Δx B = B −1 A q x q subject to x B − Δx B ≥ 0. Let d = B −1 A q. If d ≤ 0, no matter how much x q is increased, x B − Δx B will stay nonnegative. Hence, c T x can be arbitrarily decreased, and thus the problem is unbounded. Otherwise, select an index p = argmin 1≤i≤m {x i /d i | d i ...

  9. Duality (optimization) - Wikipedia

    en.wikipedia.org/wiki/Duality_(optimization)

    But for some classes of functions, it is possible to get an explicit formula for g(). Solving the primal and dual programs together is often easier than solving only one of them. Examples are linear programming and quadratic programming. A better and more general approach to duality is provided by Fenchel's duality theorem. [18]: Sub.3.3.1