Consider the following linear program $\min c^Tx$ s.t. $Ax\leq b$, where $A\in\mathbb{R}^{m\times n}$ We know that the optimal solutions $x^*$ of the above linear program are not unique. And we also know that for an optimal solution $x^*$, some constraints are binding, defined as $B=\{i\in[m]|A_ix=b_i\}$, where $A_i$ is the $i$th row of matrix A. My question is that is the set of binding constraints $B$ unique for a given linear program? how to prove that or disapprove that?
Is the set of binding constrains of linear programming unique?
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linear-algebra
optimization
convex-analysis
linear-programming
1 Answers
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Different basic feasible solutions will have different sets of binding constraints. That is, a basic feasible solution is determined by the set of binding constraints for it.
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0Does there exists two optimal solutions with different sets of binding constraints? – 2017-03-01
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0If there is more than one optimal solution and at least one of them is **basic**, then yes. That basic optimal solution is determined by its set of binding constraints, and for any other optimal solution at least one of those must be non-binding. On the other hand, there are examples with no basic optimal solutions. For example, consider $\min x_1 + x_2$ s.t. $x_1 + x_2 \le 0$. The optimal solutions are $x_1 = -x_2$, $x_2$ arbitrary, and each of these have the one constraint $x_1 + x_2 \le 0$ binding. – 2017-03-01
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0Thanks! Does this still hold for quadratic program with linear constraints? – 2017-03-01