Table of Contents

## What is optimality gap?

Generally the difference between a best known solution, e.g. the incumbent solution in mixed integer programming, and a value that bounds the best possible solution.

## What is duality gap in optimization?

In optimization problems in applied mathematics, the duality gap is the difference between the primal and dual solutions. If is the optimal dual value and is the optimal primal value then the duality gap is equal to. . This value is always greater than or equal to 0 (for minimization problems).

**What is dual solution?**

In mathematical optimization theory, duality or the duality principle is the principle that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. If the primal is a minimization problem then the dual is a maximization problem (and vice-versa).

**What is optimality gap in cplex?**

The optimality is proven if the upper bound and the lower bound evaluate the same value, i.e. CPLEX could prove an optimality gap of 0%. Since CPLEX stops with a solution that has a gap of 0.57%, I would assume that you configured an MIP-gap <1%.

### Why is duality gap important?

The duality principle provides that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. The solution to the dual problem provides a lower bound to the solution of the primal (minimization) problem.

### What is integrality gap?

Integrality gaps essentially represent the inherent limits of a particular linear or convex relaxation in approximating an integer program. Generally, if the integrality gap of a particular relaxation is x, then any approximation algorithm based on that relaxation cannot hope to do better than an x-approximation.

**What is LPP?**

Linear Programming Problems (LPP): Linear programming or linear optimization is a process which takes into consideration certain linear relationships to obtain the best possible solution to a mathematical model. It is also denoted as LPP.

**What is the difference between primal simplex and dual simplex?**

The basic difference between the regular Simplex Method and the Dual Simplex Method is that whereas the regular Simplex Method starts with basic feasible solution, which is not optimal and it works towards optimality, the dual Simplex Method starts with an infeasible solution which is optimal and works towards …

## What is primal LP?

The primal LP is defined by: A set of n variables: . For each variable , a sign constraint – it should be either non-negative ( ), or non-positive ( ), or unconstrained ( ).

## What is duality principle?

duality, in mathematics, principle whereby one true statement can be obtained from another by merely interchanging two words. It is a property belonging to the branch of algebra known as lattice theory, which is involved with the concepts of order and structure common to different mathematical systems.

**What is the duality theorem?**

The duality theorem states that: • if the primal problem has an optimal solution, then so has the dual, and zP = zD; 1 Page 2 • if the primal problem is unbounded, then the dual is infeasible; • if the primal problem is infeasible, then the dual is either infeasible or unbounded.

**What does integrality mean?**

the state of being total and complete

Definitions of integrality. the state of being total and complete. synonyms: entireness, entirety, totality. types: full treatment, kit and boodle, kit and caboodle, whole caboodle, whole kit, whole kit and boodle, whole kit and caboodle, whole shebang, whole works, works.

### What are integrality constraints?

The integrality constraints allow MIP models to capture the discrete nature of some decisions. For example, a variable whose values are restricted to 0 or 1, called a binary variable, can be used to decide whether or not some action is taken, such as building a warehouse or purchasing a new machine.

### What is optimal solution in LPP?

Any point in the feasible region of a linear programming problem that gives the optimal value (maximum or minimum) of the objective function is called an optimal (feasible) solution.

**What is an optimum solution?**

An optimal solution is a feasible solution where the objective function reaches its maximum (or minimum) value, that means the maximum profit or the least cost.

**What is the difference between big M method and two phase method?**

The Big M technique is a rendition of the Simplex Algorithm that first tracks down a best practical arrangement by adding “counterfeit” factors to the issue. In Two Phase Method, the entire strategy of taking care of a straight programming issue (LPP) including fake factors is isolated into two stages.

## What is the difference between simplex method and revised simplex method?

The revised simplex method is mathematically equivalent to the standard simplex method but differs in implementation. Instead of maintaining a tableau which explicitly represents the constraints adjusted to a set of basic variables, it maintains a representation of a basis of the matrix representing the constraints.

## What is dual feasible?

More generally, define a tableau to be dual feasible if every reduced cost has the right sign for optimality. That is, when we’re solving a minimization problem, a dual feasible tableau is one with nonnegative reduced costs.

**What is duality theorem?**

**What is the duality gap in optimization?**

In optimization problems in applied mathematics, the duality gap is the difference between the primal and dual solutions. If . This value is always greater than or equal to 0 (for minimization problems). The duality gap is zero if and only if strong duality holds. Otherwise the gap is strictly positive and weak duality holds. .

### What is G and I optimality?

A popular criterion is G-optimality, which seeks to minimize the maximum entry in the diagonal of the hat matrix X (X’X) −1 X’. This has the effect of minimizing the maximum variance of the predicted values. A second criterion on prediction variance is I-optimality, which seeks to minimize the average prediction variance over the design space.

### Does optimality theory explain the opacity of surface form?

The opacity of such phenomena finds no straightforward explanation in Optimality Theory, since theoretical intermediate forms are not accessible (constraints refer only to the surface form and/or the underlying form).

**What is an optimality criteria?**

In statistics, an optimality criterion provides a measure of the fit of the data to a given hypothesis, to aid in model selection.