Optimal solutions with a bounded number of non-zero variables
Consider the linear program in equational form:
begin{align}
text{maximize} && c^T x\
text{subject to} && A x = b , x geq 0
end{align}
It is known that, if there are $m$ equational constraints (i.e. the matrix $A$ has $m$ rows), and the problem has an optimal solution, then it has an optimal solution in which at most $m$ variables are non-zero; it is called an optimal basic feasible solution.
MY QUESTION IS: is there a larger family of optimization problems (not necessarily linear) with the same property?
I.e., is there a family of optimization problems, that always have a solution in which the number of non-zero variables is bounded by some function of the number of constraints?
optimization convex-optimization linear-programming nonlinear-optimization
asked yesterday
Erel Segal-Halevi
4,24111760
add a comment |
Consider the linear program in equational form:
begin{align}
text{maximize} && c^T x\
text{subject to} && A x = b , x geq 0
end{align}
It is known that, if there are $m$ equational constraints (i.e. the matrix $A$ has $m$ rows), and the problem has an optimal solution, then it has an optimal solution in which at most $m$ variables are non-zero; it is called an optimal basic feasible solution.
MY QUESTION IS: is there a larger family of optimization problems (not necessarily linear) with the same property?
I.e., is there a family of optimization problems, that always have a solution in which the number of non-zero variables is bounded by some function of the number of constraints?
optimization convex-optimization linear-programming nonlinear-optimization
asked yesterday
Erel Segal-Halevi
4,24111760
I don't think so.
– LinAlg
yesterday
add a comment |
Consider the linear program in equational form:
begin{align}
text{maximize} && c^T x\
text{subject to} && A x = b , x geq 0
end{align}
It is known that, if there are $m$ equational constraints (i.e. the matrix $A$ has $m$ rows), and the problem has an optimal solution, then it has an optimal solution in which at most $m$ variables are non-zero; it is called an optimal basic feasible solution.
MY QUESTION IS: is there a larger family of optimization problems (not necessarily linear) with the same property?
I.e., is there a family of optimization problems, that always have a solution in which the number of non-zero variables is bounded by some function of the number of constraints?
optimization convex-optimization linear-programming nonlinear-optimization
asked yesterday
Erel Segal-Halevi
4,24111760
Consider the linear program in equational form:
begin{align}
text{maximize} && c^T x\
text{subject to} && A x = b , x geq 0
end{align}
It is known that, if there are $m$ equational constraints (i.e. the matrix $A$ has $m$ rows), and the problem has an optimal solution, then it has an optimal solution in which at most $m$ variables are non-zero; it is called an optimal basic feasible solution.
MY QUESTION IS: is there a larger family of optimization problems (not necessarily linear) with the same property?
I.e., is there a family of optimization problems, that always have a solution in which the number of non-zero variables is bounded by some function of the number of constraints?
optimization convex-optimization linear-programming nonlinear-optimization
optimization convex-optimization linear-programming nonlinear-optimization
asked yesterday
Erel Segal-Halevi
4,24111760
asked yesterday
Erel Segal-Halevi
4,24111760
asked yesterday
Erel Segal-Halevi
4,24111760
asked yesterday
Erel Segal-Halevi
4,24111760
asked yesterday
Erel Segal-Halevi
4,24111760
4,24111760
I don't think so.
– LinAlg
yesterday
add a comment |
I don't think so.
– LinAlg
yesterday
I don't think so.
– LinAlg
yesterday
I don't think so.
– LinAlg
yesterday
add a comment |
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I don't think so.
– LinAlg
yesterday