Questions tagged [dual]

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Dual to Primal conversion

I recently tried to solve a primal minimization problem using its maximization dual. In the optimal simplex tableau of the dual, there was a slack variable and only one dual variable in the basis. So, ...
1 vote
71 views

Termination criterion for the Phase 1 algorithm in column generation

To initialize column generation, I obtain a feasible set of columns via the following Phase 1 algorithm. The restricted master problem for this algorithm is formulated as follows. min ...
• 615
1 vote
83 views

The dual values and change in the variables values

For a constraint as Ax <= b, the dual shows the change in the objective function if the RHS increases by 1 unit. Now my question is that how we can determine how the optimal values will change by 1 ...
• 400
119 views

Dual of a LP with a variable restricted to be negative

I have to find the dual of following LP: Minimize $x_1 + x_2 + x_3$ subject to $4x_1 + 8x_2 \geq3$ $7x_2 + 4x_3 \leq 6$ $3x_1 - 2x_2 + 5x_3 = 7$ $x_1 \leq 0, x_2 \geq 0, x_3$ is unrestricted I ...
• 1
397 views

Understanding reduced costs and dual values

I have a headache regarding calculating the reduced costs of a linear program. I am using Pulp for modelling and CBC for solving. What I do is, I model a linear programming problem based on a rhs ...
• 1,143
134 views

How to find robust counterpart of sum of logit functions?

Suppose function $\mu_i(y):\mathbb{R} \rightarrow \mathbb{R}$ is a logit function, $\mu_i(y)=1/(1+\exp(-y))$. Also, we assume that $\mathbf{x}_i\in \mathbb{R}^d$ and $\theta \in \mathbb{R}^d$. I am ...
• 2,160
364 views

CPLEX stuck in solve method - dual simplex solved model

I am solving a (rather large) MILP in CPLEX in Java. I also wrote a solution extractor method, which works most of the time. However, when CPLEX tries to solve my problem with the dual simplex method ...
• 29
1 vote
67 views

Developing the dual model

I have the following linear program: $$\min \sum_{\vec{s} \in S} \sum_{\vec{z} \in \Xi_{\vec{s}}} c(\vec{s}, \vec{z}) \times \chi(\vec{s}, \vec{z}) \label{eqADP4}$$ \...
• 615
1 vote
67 views

This is my model. \begin{align} \min_x&\quad\sum_{e\in E} X_e p_e \\ \text{s.t.}&\quad\sum_{e \in E: T(e)=i} X_e - \sum_{e \in E: H(e)=i} X_e = \begin{cases}1, \;\text{if}\;i=s\\-1,\;\text{if}...
• 517
1 vote
157 views

Help with dual of a problem

Could anyone confirm me if I write the correct dual for my problem? The different sets confuse me a lot. $s$ is the source node and $t$ the sink node. I'm uncertain if the last dual constraint in ...
• 517
66 views

Dual of the alternative solutions

Suppose we have two alternative solutions for a linear program. Are their corresponding dual solutions the same? (in terms of the values for each dual variable)
• 400
333 views

Finding the dual problem of a minimum problem

How to convert the following primal problem into its dual problem: \begin{align} \min_{x,z}&\quad a^\top x + b^\top z\\\text{s.t.}&\quad Ax-d \le Cz \\&\quad x\ge 0, z \le 0. \end{align} I ...
• 131
239 views

Electricity market clearing price using fixed-MIP formulation?

Dual information of electricity markets clearing problem is required to calculate the marginal clearing price. As most electricity market problems are based on MIP (and dual information of MIP is not ...
• 189
2k views

Is optimal solution to dual not unique if optimal solution to the primal is degenerate?

If optimal solution to the primal is degenerate, does it necessarily follow that optimal solution to dual not unique? That is, is uniqueness an unnecessary assumption? Spin-off from here. In my ...
• 59
422 views

Derive "true" shadow price for degenerated LPs using commercial solvers (e.g. Gurobi)

In linear programming for an optimal primal degenerate solution the values of the dual variables are in general not identical with the corresponding shadow prices. Several proposals on how to find the ...
• 93
70 views

Deriving KKT Conditions for time-step equations

I have a variable $e(h)$, and below is the part of the Lagrangian equation where I am taking the derivative with respect to $e(h)$. \frac{\partial }{\partial e(h)} \hspace{.2cm}\mu_1(h)(e(h)-\bar{E}...
• 803
1k views

Physical Interpretation of a dual of an LP

I was recently asked to physically interpret a dual of an LP for an audience who does not know mathematics/OR (without LP, dual, bounds, etc.). Though I attempted it and was very close to what the ...
146 views

On dual-formulation of a given primal for a set-covering problem

I need to solve an LP-relaxation of an airline crew pairing optimization problem (CPOP). The problem formulation is a modified SCP and is as follows: Primal of the CPOP: \begin{align}\min&\quad\...
152 views

Dual variables associated with same equation for different time instants

I have three equations that are essentially the same equation defined for three time instants. The equations are basically calculating the state of energy of an energy storage facility. \begin{align} ...
• 803
46 views

Derivations for two formulae for obtaining optimal dual variable values from the optimal primal tableau

We're being taught Industrial Engineering and Operations Research for the first time this semester. Referring to the book by Hamdy A. Taha, I noticed the mention of two formulae for swiftly obtaining ...
90 views

KKT conditions validation- one dual variable equating to two values

I have the following optimization problem: \begin{alignat}2\min &\quad A(t)\cdot x(t)-B(t)\cdot y(t)+C(t)\cdot z(t)-D(t)\cdot k(t)\\\text{s.t.}&\quad z(t)+z_1(t)-y(t)-y_1(t)+x(t) = k(t);&...
• 803
178 views

Finding Dual Objective

I have the following simplified optimization problem: \begin{align}\max &\quad ax+by\\\text{s.t.}&\quad0 \le x \le \overline X\\&\quad0 \le y \le\overline Y\\&\quad z = E-x+\beta\cdot ...
• 803
1k views

Column generation stabilisation

Has anyone performed a benchmark of the various stabilisation techniques in column generation? Which ones perform better for set pack/partition/cover problems like VRP? And are there theoretical ...
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