A non-convex optimization problem is one where either the objective function is non-convex in a minimization problem (or non-concave in a maximization problem) or where the feasible region is not convex.

# Questions tagged [non-convex-optimization]

569 questions

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### Example for an unbounded sequence with an isolated cluster point with $\|x^{k+1}-x^{k}\|\to 0$

I am looking for an example of an unbounded sequence $(x^{k})_{k\in {\mathbb N}}$ such that $\|x^{k+1}-x^{k}\|\to 0$ and $(x^{k})_{k\in {\mathbb N}}$ has at least one isolated cluster point x but the sequence is not convergent. Can this happen?
Let…

ayup al

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### Characterizing (stationary) points by the number of valleys one can descent into

In non-convex optimizing of more than 2 times differentiable $f: \mathbb{R}^2 \mapsto \mathbb{R}$ we can encounter saddle points that have multiple valley one could descent into.
At $(0,0)$ there are two direction of descent: (0,-1) and (0,1).…

worldsmithhelper

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### Does every compact (not necessarily convex) set have extreme points?

Let $A$ be a set in a normed vector space. Call a point $p$ in $A$ extreme if there do not exist $p_0, p_1$ in $A$, distinct from $p$, such that $\lambda p_0+(1-\lambda)p_1=p$, for $\lambda \in [0,1]$. Can you give an example of a compact $A$ which…

user1125477

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### Which is stronger, Kurdyka-Łojasiewicz property or regular? If the comparison is not proper, what are their range of applications?

(Kurdyka-Eojasiewicz property and exponent). We say that a proper closed function $h: \mathbb{X} \rightarrow \overline{\mathbb{R}}$ satisfies the Kurdyka-Eojasiewicz $(K L)$ property at $\hat{x} \in$ dom $\partial h$ if there are $a \in(0, \infty],…

Liuwei

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### Maximizing a positive semidefinite quadratic form over the standard simplex

I am attempting to maximize a positive semidefinite quadratic form over the standard simplex.
Given a symmetric positive semidefinite (Hessian) matrix $A \in \Bbb R^{d \times d}$ and a matrix $W \in \Bbb R^{d \times n}$,
$$\begin{array}{ll}…

JJumSSu

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### Projection of an interior point of an ellipsoid onto itself

Consider
$$E := \{ x \in \Bbb R^n \mid x^T D x = 1 \}$$
an ellipsoid constructed by the diagonal matrix $D = \mbox{diag}(d_1, d_2, \dots, d_n)$ with $d_i > 0,\ \forall i \in [n]$. Suppose that $z$ is inside the ellipsoid, $z^T D z < 1$. What is the…

Sam

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### Minimize $\left\lVert \mathbf{x} \right\rVert_1 - \left\lVert \mathbf{x} - \overline x \,\mathbf{1}_n\right\rVert_1$ over the unit $n$-cube

I want to solve the following minimization problem
$$\min_{\mathbf{x} \in [0, 1]^n} \left\lVert \mathbf{x} \right\rVert_1 - \left\lVert \mathbf{x} - \overline x \,\mathbf{1}_n\right\rVert_1$$
where $\mathbf{x} = (x_1,x_2,\dots,x_n)$ and…

huangbiubiu

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### On duality gap in general non-convex problems

I am trying to solve a non-convex constrained minimization problem. From my understanding, the dual function is always concave and provides a lower-bound to the primal problem for any dual variable(s). Since it is concave, (if it is nice enough) I…

Teodorism

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### Is there a second-order conic relaxation method for the bilinear term $z=xy$?

I hope to find a second-order conic (SOC) relaxation for $z = xy$, but it seems very hard.

Shuai Lu

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### Unobserved values in SVD and matrix factorization problems

In class, in the context of matrix completion (recommender Systems, collaborative filtering), we have seen that the Eckart-Young theorem says the best rank-$k$ approximation to a matrix $A$ is given by taking the dominant first $k < \mbox{rank}(A)$…

gammaALpha

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### Least squares problem with a quadratic constraint

I have to minimize $e_1^2+e_2^2+e_3^2$
subject to
$\mathbf{e}^\top \mathbf{A} \mathbf{e} + \mathbf{e}^\top \mathbf{b} +c =0$
with $\mathbf{e} = [e_1, e_2, e_3]^\top$
I know that matrix $\mathbf{A}$ is positive semidefinite.

JavierG

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### Minimize $ \mbox{tr} ( X^T A X ) + \lambda \mbox{tr} ( X^T B ) $ subject to $ X^T X = I $ - Linear Matrix Function with Norm Equality Constraint

We have the following optimization problem in tall matrix $X \in\mathbb R^{n \times k}$
$$\begin{array}{ll} \text{minimize} & \mbox{tr}(X^T A X) + \lambda \,\mbox{tr}(X^T B)\\ \text{subject to} & X^T X = I_k\end{array}$$
where $A \in \mathbb R^{n…

E.J.

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### How to solve this polynomial optimization problem using KKT conditions?

I am trying to find a general recipe for a nonlinear constrained optimization problems. Please correct where wrong or where parts are missing.
$$
\begin{aligned}
&\min &f(x,y)&=xy+y^3\\
&\text{such that }&x &\geq y^2\\
&&x+y&\leq…

zohaad

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### Can an optimization problem with constraints of the form $x_i \leq x_j x_k$ be made convex?

I am working on a research project with an optimization problem of the form
$$\begin{array}{ll} \text{minimize} & c^T x\\ \text{subject to} & A x + b = 0\\ & x_i \le x_j x_k\\ & x \in \mathbb R_+^n\end{array}$$
where $x_i$, $x_j$ and $x_k$ are…

Egill Juliusson

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### Finding maximum of function

I'm struggling to find the maximum of this function $f:\mathbb{R}^n\times\mathbb{R}^n \rightarrow \mathbb{R}$
$$ f(x,y) = \frac{n+1}{2} \sum_{i=1}^n x_i\,y_i - \sum_{i=1}^n x_i \sum_{i=1}^n y_i,$$
where $x_i,y_i\in[0,1]$ for $i=1,...,n$. It reminded…

user480959

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