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# Questions tagged [convex-optimization]

A convex optimization problem consists of either minimizing a convex objective or maximizing a concave objective over a convex feasible region.

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### Checking for local minimizer on boundary point

My specific question: For a convex, second-order continuously differentiable function, $f:S_{+}^{n} \to \mathbb{R}$, where $S^{n}_{+}$ denote the set of positive semidefinite matrices, how can we ...
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### Artin's vanishing theorem: a line restricts to a circle in order to calculate critical points

Two different ways of calculating critical points are given from the formulas (I hand wrote them at a class so it's not 100% the formulas but they're close): $(-1)^{dim} \mathcal{X}(X)$ = # of ...
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### Solving a constrained optimisation problem where the algebra doesn't make sense

I have the following function to maximise with respect to $x$, $p$, and $\eta$: $\frac{p\eta}{R(x)}[v(x)-\bar\theta R(x)]+(1-\frac{p\eta}{R(x)})p\eta-(1+\lambda)tpx-\lambda R(x) (1)$ subject to the ...
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### Calculating the Linear Transformation of a Norm-Bounded Convex Set

I am working on implementing mechanism 1 from this paper. I have limited knowledge on convex optimization and am not sure how to derive an explicit expression for the linear transformation of the unit ...
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### Gradient descent over a restricted convex domain — how do we guarantee that we stay in the domain if the global minimizer is outside of it?

Let $f:\mathbb R^d \rightarrow \mathbb R$ be a convex function and $A \subset \mathbb R^d$ a convex set. We are interested in finding the minimum of $f$ over $A$. We have the gradient of $f$ and we ...
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### Convexity of general quadratic function [closed]

Let $f(x) := x^T A x + b^T x + c$. If we only know $A \in \mathbb{R}^{n \times n}$, why does convexity of $f$ require that $A + A^T$ be positive semidefinite (PSD) and why does strong convexity ...
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### Lagrangian function and first order necessary optimality conditions

I am given the following equality constrained convex QP. $$\min_x \frac{1}{2}x'Hx+g'x$$ $$st. A'x+b=0$$ with $H\succ 0$. I want to find the Lagrangian function for this problem and the first order ...
### Proximal operator of squared $\ell_1$-norm
For any $a \in \mathbb R^d$ and $t \ge 0$, let $p_t(a)$ be the unique minimizer of $f_t(x;a) := \|x-a\|_2^2 + t\|x\|_1^2$ over $x \in \mathbb R^d$. Question. Is there an analytic formula for $p_t(a)$ ?...
Consider the unconstrained non-convex optimization problem: $$\min\limits_{x,y} f(x,y)$$ Suppose that for fixed $x$, the function $y \mapsto f(x,y)$ is convex. In this case, I believe there are two ...