# Questions tagged [non-convex-optimization]

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### Under what condition can we prove $\nabla_x \min_y f(x,y)=\nabla_x f(x,y^*)$ where $y^*=\arg\min_y f(x,y)$?

Let $f: \mathbb R^n\times \mathbb R^m\to \mathbb R$ be a function. I wonder under what condition can we prove $\nabla_x \min_y f(x,y)=\nabla_x f(x,y^*)$ where $y^*=\arg\min_y f(x,y)$. For example, ...
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### Understanding non-convex subgradients and normal cones

I think I have a very good understanding of subgradients of convex functions and normal cones to convex sets. On the other hand, I have a lot of difficulties understanding them in the non-convex setup....
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### Why the result of the non-convex optimization problem will be farther and farther away from the optimal

When I try to solve a optimization problem by Riemannian stochastic variance reduced gradient algorithm(RSVRG), the formulation of problem like $\frac{1}{N}\sum_{i=1}^Nf_i(x)$ and $f_i(x)$ is a non-...
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### Hardness of concave minimization problem

I have an optimization problem $\underset{x}{\min} ~ c(x) - k \cdot x$ where $c(x)$ is a non-decreasing concave function with $c(0) = 0$, $x \in C \subseteq \mathbb{R}^d_{\geq 0}$. By non-decreasing, ...
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### Variational forms of non-convex functions

I am trying to understand what kind of variational forms exist for non-convex functions. Alternatively, are there conjugate forms which attain strong duality? For a non-convex function $f$, I am ...
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### Necessary optimality condition for quadratic programming: a solution of a constrained QAP is a solution of a LP

I have a concern about a result given by Murty in  and also written by Floudas and Visweswaran in  They consider a QP: \begin{array}{ll}{\min _{x} Q(x)} & {=c^{T} x+\frac{1}{2} x^{T} D ...
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### non-convex optimization with constraint

I have a special non-convex optimization problem: $\min / \max \ f(x) + g(x) + h(x)$, subject to $| g(x) - h(x)| < \varepsilon$, where $f(x)$ is non-convex, but both $g(x)$ and $h(x)$ are ...
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### Why to multiply the penalty by $n$ in the penalized least squares and likelihood?

In the SCAD paper by Fan and Li (2001), there exist two forms of penalized least squares as follows: $$\frac{1}{2}\left \| y-X\beta \right \|^2+\lambda \sum_{j=1}^{d}p_j (\left | \beta _j \right |),$$ ...
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### Are there any solvers to Chance Constrained Programming Problems

I'm trying to solve a chance constrained programming (CCP) problem $\min_x f_0(x, \xi), \text{ such that } \mathbb{P} ( f_i(x, \xi) \ge \alpha_i ) \le \epsilon_i, \text{ where } i = 1,2,\cdots, m$ ...
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### Non-convex quadratic optimization

I would like to optimize the following system: $$\min_{q,\|q\|=1} \sum_i^n |q^T M_i q|$$ More details: the size of the unknown vector $q$ is $4 \times 1$, $M_i$ is a matrix of size $4\times 4$. It ...
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### Projected gradient descent for non-convex optimization problems

My question is in regards to the minimization of a convex function where the feasible set of solutions is non-convex. Can projected gradient descent (PGD) be used here to obtain a stationary solution? ...
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### Is all non-convex optimization heuristic?

Convex Optimization is a mathematically rigorous and well-studied field. In linear programming a whole host of tractable methods give your global optimums in lightning fast times. Quadratic ...