# Questions tagged [convex-optimization]

Optimization with convex constraints and convex objectives; notions related to convex optimization such as sub-gradients, normal cones, separating hyperplanes

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### Minimum mean over all random variables subject to logarithm constraint

Does the following problem have a solution? $$\min_X \mathbb{E} X \quad\text{subject to}\quad \mathbb{E} \log X = C.$$ Here, the minimization is with respect to all integrable random variables $X$ ...
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### Minimize a function through its upper approximation

I have some function $f(x) : \mathbb{R}^n \to \mathbb{R}$ (n is about several thousands, say $1000 \leq n \leq 10000$) to minimize over some constraints $g(x) \leq 0$ (by the way, $g$ is quadratic ...
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### Minimization of a discrete valued function

$$\min_{f} \sum_{i=1}^n \max \left( 0, 2f(i) - f(i-1) -f(i+1)\right),$$ where the minimum is taken over all the functions $f$ from $\{0,1,2,\ldots,n+1\}$ to $[0,x]$, $x <1$, such that $f$ is non-...
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### Zeroth order method with near-optimal rate that works in practice?

I want to find a ZO (zeroth-order, i.e. no access to gradient) algorithm to minimize a strongly-convex deterministic objective (say, as a sum of smooth and nonsmooth proximable functions). I want such ...
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### Minimax optimization of diagonal entries of function of matrix

Let $\mathbf{A}$ and $\mathbf{U}$ be arbitrary complex $M\times N$ and $N\times M$ matrices, respectively. Let denote superscript $(\cdot)^{\dagger}$ and $(\cdot)^{\mathrm{H}}$ as pseudo-inverse and ...
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### Matrix Completion SDP Relaxation and Duality

I am studying the matrix Completion problem, as well as its SDP relaxation. However, I am having trouble deriving the final SDP form of the matrix completion problem. I will give some background, ...
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### Showing existence of a solution to an underdetermined system of equations with non-negativity constraints

Let $K$ be a positive integer, let $p\in (0,1)$, and let $\{W(k,i),W^B(k,i), \varphi_k(i)\}_{1\leq i\leq k\leq K}$ be variables. I need to prove that there exists a solution to the following system ...
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Let $f:R\rightarrow R$ be a concave function with a unique and finite maximum. Let $$g(x, \beta) = \beta f(\alpha \cdot x) + (1-\beta) f((1-\alpha) \cdot x),$$ where $\alpha \in [0,1/2]$ and $\beta \... 0answers 54 views ### subgradient in a predual under weak* continuity Let$X$be a Banach space. Suppose$f:X^*\to\mathbb R\cup\{\infty\}$is convex, has closed and bounded (and so weak*-compact) effective domain, and is weak*-continuous on its effective domain. In ... 0answers 94 views ### How are the$L^2$and$\sup$norms related on the space of strongly convex functions? Given a convex compact set$X \subset \mathbb R^d$with interior containing the orign let$V$be the space of all smooth functions$f: X \to \mathbb Rwith the properties: The function is strongly ... 0answers 39 views ### Strict complementarity for quadratic programming Consider the quadratic program \begin{align*} \text{min}_{x\in \mathbb{R}^n} \ &\tfrac{1}{2}x^THx + f^Tx\\ \text{st.} \ & Ax \leq a \\ & Cx = c \\ \end{align*} for matricesA\in \mathbb{R}...
Let $X$ be a locally convex space, $D \subseteq X$ a nonempty compact convex set, and $f: D\to\mathbb R$ a continuous convex function. Question: Is there any known, interesting, alternative ...
Let $(E,\mathcal E,\lambda)$ be a measure space; $f:E\to[0,\infty)^3$ be $\mathcal E$-measurable with $\|f\|\in\mathcal L^2(\lambda)$; $\tilde p:=\alpha_1f_1+\alpha_2f_2+\alpha_3f_3$ for some \$\...