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0
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1answer
121 views

Minimal point of the intersection of convex sets.

I am trying to find out if there is any known result in convex optimization that implies the following statement: "A minimal point of the intersection of $N > 2$ convex sets in $\mathbb{R}^2$ is ...
0
votes
1answer
160 views

A certain type of quadratic problem.

I am interested in solving the following equality constrained quadratic (?) problem. \begin{align} \min_{u^{H}u=1}~(u^{H}A_1u) \\\ s.t.~ u^{H}A_2u=0 \end{align} $A_1$ and $A_2$ are $N\times N$ ...
2
votes
2answers
223 views

A certain type of Quadratic Constrained Quadratic Programming Problem (QCQP)

Let $P_1$, $P_2$ be two hermitian matrices. Can anyone comment about the following (QCQP) \begin{equation} \min_{z}~z^{H}z \\\ ~~subject~to ~z^{H}P_1z+1\leq 0,~z^{H}P_2z+1\leq 0 \end{equation} I am ...
0
votes
2answers
257 views

Is minimum of convex envelope the same as minimum of the original function?

Hello everyone my question is: $Question:$ Consider a function $f:X \rightarrow \mathbf R$ where $X$ is a convex subset of $\mathbf{R}^n$. The convex envelope of $f$ over $X$ is defined as the ...
1
vote
0answers
340 views

Definition and Convergence of Iteratively Reweighted Least Squares

I've been using iteratively reweighted least squares (IRLS) to minimize functions of the following form, $J(m) = \sum_{i=1}^{N} \rho \left(\left| x_i - m \right|\right)$ where $N$ is the number of ...
0
votes
1answer
98 views

Is these two optimization problems share the same solution?

Hello all, I am dealing with some SDP optimization, and I come across the following problem. The optimization problem is given by \begin{aligned} &\operatorname*{min}_{t_1,\ldots,t_m,X}\ \sum ...
1
vote
1answer
228 views

Question regard checking convexity by “restriction to any line that intersects the function domain”

Hello all, I have a question (probably stupid one) about the fact that " A function is convex if and only if it is convex when restricted to any line that intersects its domain". In Stephen Boyd and ...
0
votes
1answer
619 views

Is a jointly convex function of x and y convex as a function of x when y=z(x)?

Hi, Suppose that $x \in R^m, y \in R^n, z(x) \in R^n$, and $f(x,y)$ is convex in $(x,y)$. Is $f(x,z(x))$ a convex function in $x$ for arbitrary continuous functions $z(x)$? Thanks!
2
votes
1answer
275 views

Proving that a specific function is quasiconvex

Hello all, Assume we have a sequence of quasiconcave functions (in $X$) denoted by $f_{i,j}(X)$ for $i,j = 1,\ldots,n$. Denote by $F(X)$ the $n\times n$ matrix whose $(i,j)$ entry is the function ...
2
votes
2answers
346 views

Does the minima of a sequence of convex convergent functions converge?

Suppose $f_1,f_2,\ldots $ is a sequence of convex functions that converges to a continuous convex $f$. Let $x_1^*,x_2^*$ be their respective (not necessarily unique) minima, and let y be a minima of ...
3
votes
0answers
143 views

Ways to establish equality of measures on locally compact spaces

Let $M$ be a locally compact space and $\mu$ be some probability measure on $M$. Let $y^\ast \in M$, $f(x,y)$ be a real continuous bounded function $M \times M \to \mathbb{R}$. Consider an equality $$ ...
0
votes
0answers
141 views

On the remarkable property of the concave function's level sets

Consider a smooth function $f(x) \colon \mathbb{R}^2_+ \to \mathbb{R}_+$ such that $f$ is concave and positively homogeneous of order one. Consider a linear transform $P$ given by matrix $$ P = ...
2
votes
1answer
965 views

Sufficient conditions for gradient descent convergence

I have an unconstrained optimisation problem with convex objective function $f(x)$. Suppose I have access only to some function of the gradient $\hat{\nabla}= g(\nabla f)$, and I take gradient steps ...
2
votes
1answer
581 views

Projection exists => Uniformly convex?

Hello, I know that: Let X be a uniformly convex Banach-Space, $a\in X$ and $C\subset X$ closed and convex, then there is a unique $b\in C$ with $\left\Vert a-b\right\Vert=\inf_{x\in C}\left\Vert a-x ...
0
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0answers
116 views

variational problem under convexity constraints

I wonder if there is any method to compute variational problems subject to certain shape constraints (e.g., convexity, monotonicity, etc.). The literature I found on this topic (which I am no expert ...