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Questions tagged [relaxations]

A relaxation of a problem is a related problem whose solution is easier in some sense to find, while providing useful information about the solution to the original problem. One common form is a linear relaxation.

A relaxation of a problem is a related problem whose solution is easier in some sense to find, while providing useful information about the solution to the original problem.

For example, given an integer programming problem, the linear relaxation involves ignoring the restriction that solutions must be whole numbers, allowing for fractional solutions. The solution to the linear relaxation puts a bound on the objective function for the solution to the integer problem, and it may be feasible to round the fractional values in some fashion to achieve an approximately close solution.

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$n$-sphere enclosing the Birkhoff polytope

I am not a mathematician by training, so please feel free to correct my logic or descriptions when necessary: Let $P$ denote a $\textit{permutation matrix}$: $$ \begin{equation} P := \{X \in \{0,1\}^{n\times n} : X \mathbf{1}_n =…
Tolga Birdal
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Relaxation of $\min_{H} \text{tr}(H^T P H)$

Let $P \in \mathbb{R}^{N \times N}$ be a given symmetric matrix. Specially, $P$ has all zero entries on its diagonal, and all its off-diagonal entries are positive. And I want to minimize $$\begin{equation} \begin{aligned} \min_{H \in…
SouthChinaSeaPupil
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SDP relaxation of non-convex QCQP and duality gap

Short version Is there a duality gap between a QCQP problem and the SDP problem obtained through Lagrangian relaxation? A paper I'm studying is using this fact, but I cannot achieve the authors' results. Longer version I've been trying to reproduce…
user73053
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Semidefinite relaxation for Boolean least squares

I am working on the Boolean least squares problem, which comes up a lot in circuit design. In its raw form, it looks like this $$\begin{array}{ll} \text{minimize} & \operatorname{tr}(A^TAX) - 2b^TAx + b^Tb\\ \text{subject to} & X = xx^T\\ & X_{ii} =…
Eagle1992
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Is there a second-order conic relaxation method for the bilinear term $z=xy$?

I hope to find a second-order conic (SOC) relaxation for $z = xy$, but it seems very hard.
Shuai Lu
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What does it mean to dualize a constraint in the context of Lagrangian relaxation?

In the context of Lagrangian relaxation of discrete optimization problems, what does it mean to 'dualize a constraint'?
gen
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In an optimization problem, why is a rank-1 constraint non-convex?

I was studying a problem in a paper in which the author tries to use the semidefinite relaxation (SDR) technique in order to solve it. After changing the original problem using the SDR technique, a rank-1 constraint is added to the problem. The…
eHH
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What is the definition of "convex relaxation" in clustering?

I have following text from a paper i am trying to understand: I don't understand what does below sentence refers to as being convex/non-convex The problem is that even though the objectives (1) and (2) are convex the constraint that K is valid…
dariush
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Minimize $\int\limits_{0}^{1} c(t)u(t)\ \text dt$ subject to $\int\limits_{0}^{1} u(t)\ \text dt = 1$ with $∫\limits_0^1 \frac{\text dt}{c(t)} < ∞.$

I had the final exam of my (convex) optimization course last week and was completely stumped when I saw the following problem. We did not cover much of these types of problems during the course, maybe one example relating to Lagrange relaxation, so…
nazorated
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Probabilistic interpretation of optimality gap in Integer Program

Suppose I have an integer program model in the form of a minimization. I noticed that Gurobi (my solver) often finds a very good upper bound (i.e., feasible solution) whereas it takes a significant time to improve the lower bound to reduce the…
user2512443
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Convex relaxation of rank constraint

I would like to approximate a symmetric matrix $X\in \mathbb{R}^{n\times n}$ by a matrix of rank no more than $r$, i.e., a matrix $Z=EE'$ where $E\in\mathbb{R}^{n\times r}$, for some given $r\leq n$. Naturally, the following optimization problem…
Nicolas Bert Johnson
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Minimize trace($MX$) with $M$ rank-deficient and $X$ positive semidefinite

I have an optimization problem of the following form: $$\min_{X\succeq0} \mathrm{trace\;} MX$$ under the linear constraint $\mbox{diag} (X) = \mathrm{Id}$ and the non-convex constraint $\mbox{rank} (X) = 1$. The matrix $M$ is square and…
lostanlen
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Equivalence of a binary linear program and its relaxation

We know that given a binary (0-1) linear program, we can find lower/upper bounds using its relaxation. But, there are instances (such as shortest path problem with non-negative cycles, bipartite matching, max-flow, etc) for which the feasible region…
m0_as
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LP relaxation for integer linear programming (ILP)

I am familiar with LP relaxation for ILP (or IP). Assume we concern with integer minimization problem, which we formalize using ILP; we then relax the ILP into LP and we say that the LP provides a lower bound for the ILP. Why is this correct? I do…
user36774
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Showing the integrality property for an Integer Linear Program

I am trying to figure out why solving a relaxed Integer Linear Program (ILP) always give an integral solution. The ILP can be summarized as: $$\min \sum_{t\in T} \sum_{s \in S} c_s k_s^t $$ subject to: $$\begin{align} \sum_{s \in S} k_s^t = 1 \quad…
david
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