# 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.

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### Finding primal feasible solution from optimal dual

I'm reading Boyd's notes on forming the dual problem in order to decompose the primal problem. On page 4, right before the start of the next section, he talks about how given the optimal dual solution,...
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### Does strong duality hold when I dualize only a subset of the constraints?

Suppose I know that for some non-convex program: \begin{align}\min_x&\quad f(x)\\\text{s.t.}&\quad g_i(x)\leq 0, i \in C\end{align} strong duality holds for this problem. Now, suppose I form ...
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### Local optimum of dual of non-linear program

In general, suppose you have a non-convex optimization problem with constraints and you form the dual problem. If you find a local optimum for the dual problem, will the corresponding primal solution ...
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### 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 ...
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### Is the iteration-limited Simplex dual solution of a MIP node useful?

Idea Sometimes I encounter problems where Simplex spends many iterations for final convergence to the optimal objective value. Let's suppose, this happens when solving branch and bound-tree nodes as ...
38 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 ...
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### Ridge Regression lagrange duality

In every machine learning book we see that it is roughly mentioned that the ridge regression: $$p_1^* = \min\limits_{\beta} \ \left( \mathrm{RSS} + \lambda\sum_{j=1}^p \beta_j^2 \right)$$ is ...
182 views

### Can we have all reduced costs (strictly) positive?

I had a number of students claim on their homework that "All $z_j-c_j$ values are positive, therefore the solution is optimal." Of course, I noted that they should say "non-negative" instead of "...
232 views

### Dual bounds of integer programming problems

I often read in papers when branch-and-X algorithms are used to solve mixed integer programming problems, that the lower bound (in the minimization case) obtained from solving a linear programming ...
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### Recovering primal optimal solutions from dual sub gradient ascent using ergodic primal sequences

My question concerns recovering a primal optimal solution while performing dual sub gradient ascent. Denoting by $y_i$ the dual multiplier in the $i^{\text{th}}$ iteration, let \begin{equation} x_i = ...
592 views

### Correct way to get a dual extreme ray for an infeasible LP in CPLEX / C++

We are coding a Benders decomposition using CPLEX/Concert (C++) and we are having some troubles to generate a feasibility cut because we are not sure how to get an extreme ray of the dual for a primal ...
Given a conic problem $$\min \{c^\top x \mid Ax \succeq_\mathit{C} b\}$$ for an arbitrary cone $C$, how can I construct the dual to the problem? Moreover, in Linear Programming one constructs the ...