# Tagged Questions

Optimization is the process of choosing the "best" value among possible values. They are often formulated as questions on the minimization/maximization of functions, with or without constraints.

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### Invertibility of bordered Hessian

I have an optimization problem: $max_{x \in C} f(x)$ s.t. $Ax=b$, where $x \in R^n$ and $b \in R^m$, $m \le n$, adn $C$ compact. I know that $f$ is strictly quasi-concave, and that $A$ has rank $m$ (...
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### Optimization: Via manifolds point of view of Lagrange multipliers method

My basis on differential manifolds calculus and differential geometry being very superficial, I'm trying to understand this section on WP's article. I'm not being able to realize why most of the ...
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### How to prove the equivalence of these optimization problems?

I am reading some lecture notes and in one procedure step it is stated that: $$\min_{\mathbf{x}}\; \langle \mathbf{H}, \mathbf{Rx-Z}\rangle + \frac{\lambda}{2} \|\mathbf{Rx-Z}\|_F^2$$ is equivalent to ...
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### Skew-Symmetric Parts of Stochastic Matrices

It's easy to see that the set $\{W - W^T : W \in \mathbb{R}^{n \times n}\}$ is precisely the set of real skew-symmetric matrices. This continues to be the case if we restrict to (entry-wise) non-...
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### Maximizing $\|APBPA\|_2$ subject to $0 \leq P \leq I$

Given positive semidefinite matrices $A,B$, how to compute $$\max_{0 \leq P \leq I}\|APBPA\|$$ where the norm is the spectral norm, i.e., the largest singular value?
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### Coefficient variation in Objective Function in Mixed Integer programming

Assume we have the following Mixed Integer programming. MIP 1) $Z1=$ Max $Ax+By$ s.t $Cx+Dy<=E$ $x>=0$ and $y: {0,1}$ Now, assume we have the same MIP, and I just converted A to A' MIP2)...
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### Checking whether a solution to MIP is optimal

Consider a binary integer program \begin{align} \min \quad &\sum _{j \in J}f_j x_j +\sum _{i \in I} c_i y_i \notag \\ \mbox{s.t.} \quad &\sum _{j \in N_i} x_j \ge 1-y_i, \quad \forall i\in I \...
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### Find permutation matrix $X \in \{0,1\}^{N \times N}$ in order to make $XAX \geq_c B$

I need to solve a problem to find out the best permutation matrix $X \in \{0,1\}^{N \times N}$ which would maximize the number of elements in matrix $XAX$ which are above (component-wise) matrix $B$ ...
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### Prioritized solution of a linear system subject to inequality constraints

Consider the following linear system $$y = A_1 x_1 + A_2 x_2$$ subject to the linear constrains $$C_1 x_1 + C_2 x_2 \leq d$$ I am looking ...
Let's say I have the following maximization problem: $max_U{tr(AU)}$ where $A\in\mathbb{C}$ and $UU^\dagger=1$ I know that for $A\in\mathbb{R}$ and $UU^T=1$ the solution is: $U=XZ^T$ where $X$ ...