Questions tagged [duality]
For questions on duals of (primal) mathematical programs that optimize the complementary bound. When minimizing, for example, primal solutions are upper bounds, and dual solutions lower bounds on the optimal value.
75
questions
2
votes
1
answer
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Reducing optimality to feasibility in non-linear programs
It is well-known that, given a linear program: minimize $c^T x$ such that $A x\leq b$, it is possible to reduce the program to deciding feasibility of the following set of constraints:
$Ax \leq b, A^T ...
1
vote
0
answers
44
views
Distributionally Robust Stochastic Programming - Help with derivation
I've been working through this book on robust optimization of electric energy systems, and in particular chapter 4 on distributionally robust optimization. In following the derivation of section 4.2.1....
- 61
1
vote
1
answer
65
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Question about Pricing Problem in Column Generation
In column generation, we need to solve the following pricing problem :
$$\min c_j-\bf{c}^T_B\bf{B}^{-1}\bf{N}_j$$
In the book, I saw authors say that according to duality theory, $\bf{y}^T = \bf{c}^...
- 25
1
vote
1
answer
136
views
Primal-Dual Simplex Algorithm
Are there any recommended textbooks or notes to learn the details about the primal-dual simplex algorithm?
- 25
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vote
1
answer
53
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how do I find the dual when a variable has an upper bound?
What is the dual of this primal LP?
$$
max~c^Tx
$$
$$
s.t.~Ax=0
$$
$$
0 \leq x \leq d,~d \in \mathbb{R}_+^n
$$
- 892
2
votes
0
answers
62
views
Approximating an LP with an exponential number of variables and an almost-separation-oracle to its dual
Problem settings:
we have $n$ agents and a set $\mathcal{S}$ of possible world-states, where the size of $\mathcal{S}$ is exponential with respect to $n$.
Each agent $j$ has a utility function $u_j\...
- 53
1
vote
2
answers
111
views
Persisting Subproblem Infeasibility Benders Decomposition
I'm currently working on an optimization problem where I'm using Benders decomposition to solve a complex problem involving the installation of charging stations. The master problem determines the ...
- 21
2
votes
1
answer
59
views
How do you derive the Benders feasibility cuts?
starting off with a MIP that I want to solve using Benders.
so in Benders Decomposition, you add feasibility cuts in the following form:
$v^j (b - Ax) \geq 0$
with $j \in J$ being the set of extreme ...
- 139
4
votes
1
answer
68
views
Can dual values smoothing lead to generating duplicate columns?
When using smoothing for dual values stabilization in column generation, the duals used in the subproblem lie on the segment joining the stability center (inside the dual feasible region) and the ...
3
votes
1
answer
89
views
Dual norm definition: adding new constraints
For some $c >0$ and $z \in \mathbb{R}^n$, the optimal value of
\begin{align}
\begin{array}{cl}
\sup_{x \in \mathbb{R}^n}& z^\top x
\\\text{s.t.}& \lVert x \rVert \leq c
\end{array}
\end{...
- 3,980
1
vote
1
answer
114
views
Equivalent condition for indicator function
We have the following two conditions:
$C1.$ $l(x) \geq 0$ for all $x\in \mathbb{R}^n$,
$C2.$ $l(x) \geq 1$ for all $x\in \mathbb{R}^n$ such that $a+b^Tx \leq 0.$
Here $l(x) = \begin{bmatrix} x^T 1 \...
- 111
0
votes
0
answers
57
views
What the dual constraints mean?
I have a primal problem that is a network-flow:
$$
min_x \sum_{a \in A} c_a x_a \\
\text{s.t.} \sum_{a \in A:n = v} X_{a}-\sum_{a \in A:n' = v} X_{a} = \begin{cases} \;\;\;1, \; v=n^{in} \\ -1, \; v=n^...
- 507
0
votes
0
answers
38
views
Dual prices remain stable but negative in column generation
I arrive at an objective function value that I know to be optimal. However, dual prices remain negative (which leads me to believe another pattern can be added, but when I do, the dual prices don't ...
5
votes
2
answers
423
views
Simple OLS problem can only be solved in SCS. Is the dual infeasible?
Essentially, I am trying to solve a simple orthogonal least-squares (OLS) problem with some constraints — the coefficients must sum to $1$, no coefficient can be less than $0$, and no coefficient can ...
4
votes
2
answers
319
views
What happens to the dual and primal feasibility when a constraint is removed after finding an optimal solution?
Assuming I had solved the a problem to optimality, I want to remove a constraint.
What happens to primal feasibility?
What happens to dual feasibility?
How to solve this new problem efficiently?
My ...
- 253
4
votes
3
answers
308
views
Existence of extreme points in primal and dual LP
If the nonempty feasible set of a primal LP has extreme points does its dual also have extreme points? I know that a standard form LP (nonempty) always has extreme points. But I am not sure if we can ...
- 77
3
votes
2
answers
121
views
Partial Lagrangian in the Max-Flow problem
In the question:
"Partial" Lagrangian Dual in LP
It is argued that considering a partial Lagrangian $L_{partial}$, where we Dualize only some of the constraints, results in a tighter ...
- 143
5
votes
1
answer
293
views
Dantzig-Wolfe vs Benders Decomposition on the dual problem - Computational differences
My question is a follow-up to this one: Relationship between Benders’ decomposition and Dantzig-Wolfe decomposition. Here what is being discussed is the relationship between the two methods, and it is ...
- 534
1
vote
2
answers
61
views
Number of solutions to geometric program
Is it possible to determine if a Geometric Program (GP) has one, none, or infinite (primal) solutions by its structure (e.g., in terms of the number of variables, constraints, or product terms ...
- 215
3
votes
0
answers
127
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 ...
- 2,150
2
votes
1
answer
102
views
How do you recover dual variables for a minimum weight bipartite perfect matching problem?
I feel like I must be missing something obvious but this is confusing me. Let's say I have an optimal solution $x^*$ to a minimum-weight perfect bipartite matching problem on $2n$ nodes, $$\min\sum_i\...
- 163
4
votes
3
answers
1k
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How do you get the primal solution of an LP from the dual solution?
I am new to Optimization so I think the following question may be very easy, but I'm not sure how to solve it.
The dual of an LP is an LP. If we solve the dual LP, we can get the optimal value for the ...
- 141
3
votes
2
answers
264
views
Related to Lagrangian dual
In my research class our professor discuss a paper wherein the solution is obtained via a Lagrangian duality. The original problem is given below:
minimize $t$
subject to $\sum_{j \in \mathcal{M_i}}\...
- 33
4
votes
2
answers
235
views
Phase I of the simplex method and Farkas certificates
Phase I of the simplex method solves an auxiliary optimization problem to determine an initial basic feasible solution, or concludes that no such exists. Is there a way to use the solution of this ...
- 213
3
votes
1
answer
180
views
Convex optimization with linear constraints. Can I solve it analytically?
I have a constrained convex optimization problem with linear equality and inequality constraints.
\begin{align}
\label{eq:costf}
\text{minimize}\ \
&f(x_1,\dots,x_m) = \sum_{i=1}^m \frac{1}{...
- 39
1
vote
1
answer
151
views
Dual solution when solving a primal degenerate LP with the interior point algorithm
Say, we're working with an LP that is primal degenerate (optimal solution is at a vertex but with multiple bases) and not dual degenerate (optimal solution is not at a face).
If we were to solve it ...
- 173
1
vote
0
answers
65
views
Dual of a quadratic constraint
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}...
- 507
1
vote
1
answer
151
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 ...
- 507
6
votes
1
answer
137
views
Light weight proof of strong duality in linear programming
For teaching purposes, I believe it can be good to use very light-weight proofs of deep results, as it often answers the question "why is it true" better than other types of proofs. By "...
- 6,352
5
votes
0
answers
75
views
Linear programming approach to dynamic programming - an initial pair of state-decisions
I aim to solve the following Bellman equation:
\begin{equation}
v(\vec{s}) = \min_{\vec{x} \in \Xi_{\vec{s}}} \big\{c(\vec{s}, \vec{x}) + \lambda \times \sum_{\vec{s}^{'}\in S} p(\vec{s}^{'} | \...
- 615
2
votes
1
answer
174
views
What is the dual of this LP?
Here is my simple LP problem for a constant symmetric positive matrix $d$ and continuous decision variables $x$:
\begin{align}\max&\quad\sum{x_i}\\\text{s.t.}&\quad
x_i + x_j \leq d_{ij}\\&...
- 892
3
votes
1
answer
161
views
min max inside a linear program
Although this sounds like a standard minimax problem, I'm not sure how to deal with feasibility issues.
Consider a maximin linear program
\begin{align}\max_x\min_y&\quad c^\top y\\\text{s.t.}&\...
- 159
6
votes
0
answers
223
views
Airline revenue management re-solving problem
I am considering a bid prices (shadow price of the capacity constraint) problem (from Chen, L. and Homem-de Mello, T. (2009)., page 14) where the acceptable classes for booking requests for ...
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3
votes
0
answers
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
9
votes
2
answers
952
views
"Partial" Lagrangian Dual in LP
Consider the optimization problem
\begin{align}\label{opt-lp}\tag{Primal}
\begin{array}{cl}
\underset{x \in \mathbb{R}^n}{\text{minimize}} & c^\top x \\
\text{subject to} & Ax = a \\
& Bx =...
- 3,980
4
votes
2
answers
2k
views
What is the relation between dual variables and reduced costs?
My background: Pure math current Undergrad, learned the theory of Operations Research, but pretty basic. All we covered have been dealing with problems that have only 1 constraint matrix. I have dealt ...
- 421
4
votes
1
answer
116
views
Convex-Constrained Nonconvex-Nonconcave Minimax Problem
In the mathematical optimization theory, I have taken a glance at many papers which deal with the unconstrained convex-concave or nonconvex-concave minimax optimization, i.e.,
$$
\min_{x\in X}\ \max_{...
- 155
6
votes
1
answer
396
views
Dual of a model to obtain reduced costs
I have the following model which I am going to solve with column generation.
\begin{align}
\max & \sum_{b \in B} \sum_{s \in S} \sum_{r \in \Omega_s}\beta_{bs}p_r y_{br}\label{objective-set1}\\
...
- 61
3
votes
1
answer
111
views
dual variable with constraints dealing different time limit
I am working on a unit commitment problem where I need to turn the constraints to dual. But the constraints do not deal with the same time period which makes dual process confusing. for a simple ...
- 55
7
votes
1
answer
212
views
Is there any academic reference which suggests/uses dual values as initialization of Lagrangian multipliers?
The Lagrangian relaxation approach is used to generate lower (upper) bounds for minimization (maximization) problems by moving some constraints to the objective function and multiplying them by "...
3
votes
1
answer
136
views
Method of Multipliers: Why is the next iterate always dual feasible?
I am reading this expository paper on ADMM by Boyd, et. al.
Consider the problem
\begin{align*}
&\min f(x)\\
& \ \text{s.t.} \ \ \ Ax = b
\end{align*}
with Lagrangian $L(x, \lambda) = f(x) + \...
- 193
2
votes
1
answer
520
views
Recovering Primal Solution from Dual solution
Consider the problem
\begin{align*}
&\min f(x)\\
& \ \text{s.t.} \ \ \ Ax = b
\end{align*}
In this expository paper, Boyd claims (top of page $8$) that if:
$\lambda^*$ is a dual optimal ...
- 193
2
votes
1
answer
296
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
4
votes
1
answer
228
views
Following code doesn't work in matlab with CVX
Given the following problem \begin{align}\min&\quad x_1+2x_2+3x_3+4x_4+\sum_{i=1}^4x_i\ln(x_i)\\\text{s.t.}&\quad e^\top x=1\\&\quad x\geq0\end{align}
I was asked to solved the dual ...
- 405
1
vote
0
answers
102
views
How to start the Dantzig-Wolfe decomposition?
I have the following problem: \begin{align}\min&\quad3x_1+5x_2+3x_3-2x_4+3x_5\\\text{s.t.}&\quad x_1+x_2+x_3+x_4\geq3\\&\quad3x_1+x_2+5x_3+x_4-2x_5\geq6\\&\quad x_1+2x_3-x_4\geq2\\&...
- 97
7
votes
0
answers
120
views
Estimate lagrangian multiplier based on instance characteristics
Assume we have a simple resource allocation problem, where all players have the same cost, but a different utility $a_s$. The resources assigned to a certain player must be between $L$ and $M$. ...
- 123
4
votes
1
answer
594
views
Column generation for a linear optimization problem
I have an LP that has exponentially many constraints, and just linearly many variables. The dual of the problem, therefore, has exponentially many variables, while just linearly many constraints. My ...
- 3,980
1
vote
1
answer
113
views
Find a dual problem with one dual decision variable to the problem of finding the orthogonal projection of a given vector
Given the set $T_{\alpha}=\{x\in\mathbb{R}^n:\sum x_i=1,0\leq x_i\leq \alpha\}$
For which $\alpha$ the set is non-empty?
Find a dual problem with one dual decision variable to the problem of finding
...
- 405
3
votes
1
answer
115
views
Find the dual problem of $\min_x\{||x-a_1||+||x-a_2||+||x-a_3||,a_i\in\mathbb{R}^n\}$
Find the dual problem of
$$\min_x\{||x-a_1||+||x-a_2||+||x-a_3||,a_i\in\mathbb{R}^n\}$$
I've tried the following but got stuck
$$\min_x\{||x-a_1||+||x-a_2||+||x-a_3||,a_i\in\mathbb{R}^n\}=\min_{x,z_i}...
- 405
2
votes
0
answers
63
views
Prove $\sum_{i=1}^{m}\lambda_i^*\leq\frac{f(\hat{x})-f^*}{\underset{i=1,\ldots,m}{\min}(-g_i(\hat{x}))}$
Consider the primal problem \begin{align}f^*=\min&\quad f(x)\\\text{s.t.}&\quad g_i(x)\le0\tag P\end{align} where $f,g_i$ are convex functions. Suppose there exists $\hat{x}$ such that $g_i(\...
- 405