Questions tagged [relaxation]

For questions related to optimization problems that are obtained from other optimization problems by increasing the feasible region, typically by removing one or more constraints or changing their right-hand sides.

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5
votes
2answers
124 views

What is a general procedure to prove that the LP relaxation of an IP delivers the optimal IP solution?

Say that I have a binary IP $$z=\max_x \{c^\top x: Ax=b, x\in B^n\}$$ where $B^n$ is the set of $n$-dimensional $0-1$ vectors. Its LP relaxation will be $$z^{LP}=\max_x \{c^\top x: Ax=b, 0\leq x\leq 1\...
3
votes
0answers
114 views

SDP relaxation with greater-than and less-than inequalities at the same time

I am dealing with the following nonconvex fractional quadratic optimization problem \begin{align} & \min_{\boldsymbol{x}} && \max_{t \in \mathcal{T}} \frac{\boldsymbol{a}_t^T \boldsymbol{...
5
votes
1answer
74 views

Linearize a product of an integer variable (not just binary) and a continuous variable?

I have a constraint in my formulation that contains multiplication of an integer variable $y$ and a continuous variable $x$, which is $xy=q$ where $y$ is the number of units in which $q$ gets equally ...
7
votes
3answers
158 views

Ways to strengthen QCQP relaxations

I was wondering what types of methods can be used to strengthen QCQP relaxations. Our solver has all the standard stuff, like constraint propagation, presolving, etc., but some QCQP problems seem to ...
7
votes
2answers
152 views

The general meaning of “constraint relaxation” in the context of the Shortest Path Problem

I've read that in the context of the Shortest Path Problem, the use of the term "relaxation" ("relaxing edges") [...][the use of the term "relaxation"] is historical. The outcome of a relaxation ...
20
votes
5answers
271 views

Tightness of an LP relaxation without using objective function

How can we measure the tightness of a linear programming relaxation for a mixed integer linear program without using the objective value? I would like to get a measure in terms of the feasible set and ...