In an optimization problem, a slack variable is a variable that is added to an inequality constraint to transform it into an equality. Introducing a slack variable replaces an inequality constraint with an equality constraint and a non-negativity constraint on the slack variable.^[1]:131

1. Negative Slack Can Be Estimated
2. Negative Slack In Schedule
3. Negative Slack
4. Negative Slack
5. Negative Slack In Project Management

1. Slack bus – to balance the active and reactive power in the system. It is also known as the Reference Bus or the Swing Bus. The slack bus will serve as an angular reference for all other buses in the system, which is set to 0°. The voltage magnitude is also assumed to be 1 p.u. At the slack bus.
2. Negative Slack (Float) means that tasks do not start or finish in time to meet one or more critical date in the project. What Else to Consider The 14 Point Assessment tests for 'Negative Float' Negative Float: Total number of activities that are incomplete and have float less than zero working days.
3. Negative slack indicates a lack of slack over the entire project and the amount of time A) an activity must be accelerated to complete the project by the required completion time. B) an activity must be delayed to complete the project by the required completion time. C) each activity on the critical must be accelerated to complete the project by the required completion time. D) all activities.
4. There are actually two probably causes of negative Total Slack: The schedule has slipped past an inflexible constraint (such as a Must Finish On constraint). The schedule has slipped past a Deadline date. Since you say you have no constraints in the project, do you have any Deadline dates?

Slack variables are used in particular in linear programming. As with the other variables in the augmented constraints, the slack variable cannot take on negative values, as the simplex algorithm requires them to be positive or zero.^[2]

Slack variables are used in particular in linear programming. As with the other variables in the augmented constraints, the slack variable cannot take on negative values, as the simplex algorithm requires them to be positive or zero.

• If a slack variable associated with a constraint is zero at a particular candidate solution, the constraint is binding there, as the constraint restricts the possible changes from that point.
• If a slack variable is positive at a particular candidate solution, the constraint is non-binding there, as the constraint does not restrict the possible changes from that point.
• If a slack variable is negative at some point, the point is infeasible (not allowed), as it does not satisfy the constraint.

Example[edit]

By introducing the slack variable ${\displaystyle \mathbf{y}\ge \mathbf{0}}$, the inequality ${\displaystyle \mathbf{A}\mathbf{x}\le \mathbf{b}}$ can be converted to the equation ${\displaystyle \mathbf{A}\mathbf{x}+\mathbf{y}=\mathbf{b}}$.

Embedding in orthant[edit]

Slack variables give an embedding of a polytope${\displaystyle P\hookrightarrow ({\mathbf{R}}_{\ge 0}{)}^f}$ into the standard f-orthant, where f is the number of constraints (facets of the polytope). This map is one-to-one (slack variables are uniquely determined) but not onto (not all combinations can be realized), and is expressed in terms of the constraints (linear functionals, covectors).

Slack variables are dual to generalized barycentric coordinates, and, dually to generalized barycentric coordinates (which are not unique but can all be realized), are uniquely determined, but cannot all be realized.

Dually, generalized barycentric coordinates express a polytope with n vertices (dual to facets), regardless of dimension, as the image of the standard ${\displaystyle (n-1)}$-simplex, which has n vertices – the map is onto: ${\displaystyle {\mathrm{\Delta }}^{n-1}\twoheadrightarrow P,}$ and expresses points in terms of the vertices (points, vectors). The map is one-to-one if and only if the polytope is a simplex, in which case the map is an isomorphism; this corresponds to a point not having unique generalized barycentric coordinates.

References[edit]

1. ^Boyd, Stephen P.; Vandenberghe, Lieven (2004). Convex Optimization(PDF). Cambridge University Press. ISBN978-0-521-83378-3. Retrieved October 15, 2011.
2. ^Gärtner, Bernd; Matoušek, Jiří (2006). Understanding and Using Linear Programming. Berlin: Springer. ISBN3-540-30697-8.^:42

External links[edit]

Negative Slack Can Be Estimated

Negative Slack In Schedule

• Slack Variable Tutorial - Solve slack variable problems online

One of the questions that FPGA designers wonder and sometimes even argue about, is: Should the implementation tools focus on Worst Slack (WS) or Total Negative Slack (TNS)?

FPGA tools typically devote more attention to WS, but there are tradeoffs. If WS is small yet many paths fail timing, then TNS can be huge. Similarly, if TNS is slight but WS is failing by a lot, that is also a problem.

To answer this question, we added a feature to collate the data for both timing values and ran about 1000 compilations on a small design (Cyclone V, about 9% utilization) using different synthesis and fitter settings.

This is what we see. The default result is marked by the black dot. The Y-axis represents the absolute value of the TNS and the X-axis, the absolute value of failing WS.

The chart is divided into 4 quadrants, green means these compilations give better TNS and WS compared to the original design.

Negative Slack

Besides looking like a pretty smattering of colored dots, we can see that the TNS flattens out when it is closer to zero but the worst slack still keeps improving.

Negative Slack

Taking a closer look,

What do you think?
Make your own conclusions, and please share your thoughts with us!

Negative Slack In Project Management

Join Plunify Newsletter