Questions tagged [convex-optimization]
Convex optimization, a subfield of optimization, studies the problem of minimizing convex functions over convex sets. The convexity property can make optimization in some sense "easier" than the general case - for example, any local minimum must be a global minimum.
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How to handle a complex-valued equality constraint involving magnitudes in an optimization problem?
I'm working on the following optimization problem:
\begin{aligned}
\mathbf{P}: \quad
& \max_{\mathbf{z},\,\mathbf{b}} && |b_1| \\
& \text{s.t.} && C_1: |b_m| \le b_{\max}, \...
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How can I express $ y \ln\!\left(1+\frac{x}{y}\right) \ge \frac{1}{u} $ as an exponential cone constraint?
I am an engineer trying to understand how to model the following inequality in a conic-optimization-compatible form:
$
y\ln\!\left(1+\frac{x}{y}\right) \ge \frac{1}{u}, \qquad x \ge 0,\; y>0,\; u&...
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Majorize instead of maximize
In my very limited knowledge of optimization there is always an objective function to minimize or maximize.
I have a problem where there are binary and integer variables. The objective is to find a ...
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Highly Non-convex Function for Algorithm Comparison
I am designing a numerical experiment to compare the performance of various nonconvex optimization algorithms (e.g., SGD with momentum, Adam, BFGS, etc.). To do this properly, I need a challenging ...
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Optimizing for multiple inventories
I have $M$ warehouses $w_1, w_2, ..., w_m$ each with inventory on $N$ items. For example $w_1$ may have 3 of item_0, 2 of item_1, $k$ of item_n.
How does one formulate each of the following objective ...
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Are optimality guarantees useful/required from a business perspective?
Let's say you have a problem that can approached with either classical solvers that offer optimality guarantees, or some meta-heuristic or AI based approach that provides a reasonably accurate result ...
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Docplex: updating convex function via iterative solutions
Anyone with experience using docplex can give me some insight or tips regarding my situation?
I'm trying to build a convex function using matrix-vector notation. But the elements of the matrix and ...
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Invertibility of Conditional Probability Matrices in the Multinomial Model
I'm working through a proof involving the invertibility of a matrix arising from a multinomial model. This is my first time using an approach based on monomial evaluations and zero sets of polynomials ...
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Is there a convex (exponential cone?) case of one of the following problem classes?
Let $n\in\mathbb{N}$, $A\in\mathbb{R}^{n\times n}$, $b\in\mathbb{R}^n$, $c>0$, $d_1,\dots,d_{n-1}>0$ and define:
$$S:=\{x\in\mathbb{R}^n\mid Ax\le b, x_1\ge c,\dots,x_n\ge c\}.$$
Do there exist ...
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Relationship between Optimal Solutions of a Linear Program and its Integer Counterpart with Binary Constraint Matrix
Consider a matrix $\mathbf{A} \in \{0, 1\}^{K \times N}$ and a vector $\mathbf{b} \in \mathbb{Z}_{>0}^{K \times 1}$ (i.e., elements of $\mathbf{A}$ are either $0$ or $1$, and elements of $\mathbf{b}...
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Generalisation of semidefinite programming from a discrete optimsation standpoint
Semidefinte programming (with linear objectives) is sometimes explained as a generalisation of linear programming. Both approaches admit finding solutions and certificates of optimality efficiently. ...
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Property of the optimal solution of an optimization problem
$m \geq 2, n \geq 2 .[m]=\{1,2, \ldots, m\} \text { and }[n]=\{1,2, \ldots, n\}.$
Vectors $c$ $\in \mathbb{R}_{++}^{m}$(strictly positive), $w$ $\in \mathbb{R}_{++}^{n}$ and $\sum_{i \in[n]} w_{i}=1$....
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SOCP reformulation for equality constraint
I am solving the given problem using CVX. \begin{align*}
& \underset{\mathbf{v},b_{1}}{\text{max}}\left|b_{1}\right|\text{s.t} \hspace{1em}
b_{m}=\left|\sum_{p=-P}^{P}v_{p}y_{m}\left(a_{m+p}-b_{...
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Is there a convex constraint to ensure a product of stochastic matrices is lower triangular
I have an (otherwise) convex problem over two matrices, $K \in \mathbf{R}^{m \times n}$ (row stochastic) and $P \in \mathbf{R}^{n \times m}$ (column stochastic) and I would like to add an additional ...
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Epigraph Reformulation of a maximization Problem
I have a maximization problem given as:
$\underset{\mathbf{z}}{max}\hspace{1em}\left|\mathbf{z}\,^{T}\mathbf{v}\right|$, where $\mathbf{z}$ and $\mathbf{v}$ are column vectors. How can this be ...
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