Questions tagged [convex-optimization]
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97 questions
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ML Gradient Descent Convergence Theorem
Does there exist a generalization of this theorem, by Yurii Nesterov in Introductory Lectures on Convex Optimization (2004) which relaxes the assumption that the loss function is convex and shows that ...
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105
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Minimizing a submodular function containing summation and production under partition matroid constraint
I'm having difficulty solving the following problem: We're given $n$ sets $X_1,\ldots, X_n$. Each set $X_i=\{(a_i,b_i)\}$ contains poly(n) many ordered pairs of non-negatives with $0\le a_i+b_i\le 1$. ...
- 483
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Non-convex optimization with correlated minima
I am thinking of non-convex optimization problems where the minima are somehow correlated. Maybe there are symmetry relationships among minima or maybe there is regularity in spacing among minima in ...
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32
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Variants of weak optimization problems for convex sets
In their famous book, Grotschel Lovasz and Schrijver (1993) present several algorithmic problems on convex sets. Each of these problems has a strong variant and a weak variant. In particular, the ...
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cutting plane method for convex optimization
The cutting plane approach in convex optimization is a general recipe for minimizing a convex function. The argument relies on the fact that using the gradient vector, we can cut the feasible set into ...
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Convergence rates for the iterates of SGD on Lipschitz convex functions
Let $f:X \rightarrow \mathbb{R}$ be a convex and $L$-Lipschitz continuous function. Suppose $f^* = \min_{x \in X} f(x) \in \mathbb{R}$ and let $X^* = \{x \in X : f(x) = f^*\}$.
For a non-negative ...
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Deciding whether a convex region is empty
Let $S\subseteq \mathbb{R}^n$ be a convex region defined by $$g_i(x)\leq 0, ~~i\in 1,\ldots,m,$$ where $g_i$ are convex functions.
The goal is to decide whether $S$ is empty, and if not - find a point ...
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Convex optimization: is it possible to find solutions that are exactly feasible and approximately optimal in polynomial time?
In Nemirovxki's lecture notes on interior point methods, I found the following.
He defines an approximate solution as satisfying the following, for any given $\epsilon>0$:
that is: the ...
- 2,314
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Solving non-linear programming with large number of variables
Let $n \in \mathbb{N}, [n] = \{1,2,\ldots,n\}$ and consider the following optimization problem:
$$\max \sum_{i \in [n]} \sum_{j \in [n]} x_i \cdot x_j \cdot c_{i,j}$$
$$s.t.~~~~~~~~~~~~~~~~~~~~~~~~~~~~...
- 422
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Linear Programming Sensitivity to Matrix
Consider a linear program in the following standard form:
\begin{align*}
&\max c^T x &\\
&\mbox{subject to:}\\
&A x \preceq b\\
&x \succeq 0
\end{align*}
Its dual is
\begin{align*}...
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Which variant of the ellipsoid method was used for the Santa Claus problem?
As one of the steps in the article The Santa Claus problem (Bansal and Sviridenko, 2006) the following linear problem was considered (at the end of the second page, as the dual):
\begin{align*}
&\...
- 101
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What is the meaning of loss in online convex optimization?
I am studying online convex optimization, and it is stated that when we make a decision, we observe loss corresponding to our decision. In some problems like multi-armed bandit problems, we know the ...
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Can the ellipsoid method be used with a randomized separation oracle?
Suppose we are trying to solve the following optimization problem:
$$
\text{maximize } ~~ c\cdot y
\\
\text{subject to } ~~ y\in S
$$
where the region $S$ is described by an exponential number of ...
- 101
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How to prove that a given class of convex programs cannot be solved by linear programming?
Given the following program, where $f, g$ are convex functions:
$$
\text{minimize}~~ f(x)
\\
\text{subject to}~~ g(x)\leq 0
$$
the problem can be solved by convex programming algorithms, but it would ...
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Strongly polynomial time algorithm for shortest convex combination
Problem: Let $S$ be a finite set of vectors. Let $C$ be their convex hull. Compute $\operatorname{argmin}_{x \in C} \|x\|$.
Reference 1 gives an algorithm for this problem that is finite-time (Section ...
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Restriction of a convex function to {0, 1}^n
Suppose I have a real-valued convex function $f$ on the unit hypercube $[0,1]^n$, and let $\bar{f}$ be its restriction to the integer points $\{0,1\}^n$. Does $\bar{f}$ satisfy any properties, or can ...
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Maximum volume ellipsoid in an intersection of ellipsoids
Given a collection of $m$ ellipsoids in $\Bbb R^n$, compute the maximum volume ellipsoid inscribed in their intersection.
In section 8.4.2 of Boyd & Vandenberghe's Convex Optimization, this ...
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Reference showing global optimality of local minima for matrix factorization
Consider the following matrix factorization problem: Given an $n\times m$ matrix M, find $n\times r$ and $m\times r$ matrices $U$ and $V$ such that $||UV^T - M||_F^2$ is minimized.
I have heard it ...
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SVM perturbation bounds
Let $B$ be the unit ball of $\mathbb{R}^d$. Suppose that $x_1,\ldots,x_n$ are vectors in $B$ with labels $y_i\in\{-1,1\}$. We say that $w\in B$ separates this labeled set with margin $\gamma$ if $y_i(...
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program search with optimization methods for (resource bounded) Kolmogorov complexity
Are there fields of research that look at finding short programs for generating strings (therefore trying to find the (resource bounded) Kolmogorov complexity of the string), but using optimization ...
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Hessian of non differentiable convex function
The motivation of the question is the following:
Let $P$ be a set of $n$ points in $\mathbb{R}^d$. Consider the following objective(convex and differentiable) function $f:\mathbb{R}^d\rightarrow [0,\...
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A variant of randomized co-ordinate descent
Let us consider the following optimization problem.
$\mathcal{P} =\{P_1,\cdots,P_n\}$, where $P_i\subset\mathbb{R}^d$. Let $m = max_i\lvert P_i\rvert$. The goal is to find a point $c$ such that ...
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An optimization problem
I am considering the following optimization problem. Let $P$ be a set of $n$ points in $\mathbb{R}^d$
maximize $\sum_{p\in P}\vert\langle \Vert p\Vert p, \hat{x}\rangle\vert$ subject to $\Vert\hat{x}\...
- 265
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Parametrized complexity of sparse optimization
Optimization problems of the type: minimize $c^T x$ subject to [maybe some linear constraints and] $||x||_0\le k$ are known to be NP-hard. [Actually, I just realized that I don't have a reference, so ...
- 10.6k
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Tight estimates on the Lovász and Multilinear extensions of a submodular function
I assume here some familiarity with the jargon used in submodular optimization (please let me know if something is unclear).
Let $f:2^V \to \mathbb{R}$ be monotone, normalized and submodular. For ...
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Finding shortest calculation of the sum of a subset of a group, given sums for other previously summed subsets
Say $S=\{g\in G\}$ is a set of elements in an abelian group $G$ whose group operation $(+)$ is expensive to compute. Given a subset $T\subset S$, we want to compute the sum of $T$'s elements, $\...
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Gradient descent step size for strongly convex functions
Suppose we are optimizing a strongly convex function $f(x)$ via gradient descent $x_{t+1} = x_t - \eta_t \nabla f(x_t)$. By strongly convex I mean that $f(x+h) \ge f(x) + \langle \nabla f(x), h \...
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Multivariable concave function $(n - 1) f(x) >= \sum_{i=1}^{n} f(x_{-i})$
Define the multi-dimension concave function $f(x): \mathbb{R}^n_+ \rightarrow \mathbb{R}_+$ where $x \in \mathbb{R}^n_+$, here I use $\mathbb{R}_+$ to represent the range $[0, \infty)$ and we let $f(\...
user54970
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Is the Chi-square divergence a Bregman divergence?
Is the Chi-squared divergence $\sum_{i} \frac{(x(i)-y(i))^2}{x(i)}$ a Bregman divergence? I.e., can it be written as
$\phi(x) - \phi(y) - \langle\phi'(y),x-y\rangle$?
If so, what is the potential ...
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When can convex optimization be considered to be exactly solvable?
If one is trying to find the global minima of a convex function using gradient descent then one will get a run-time which is a function of $\epsilon >0$ where $\epsilon$ measures the accuracy of ...
- 1,453
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Is convex optimisation in P?
Consider a convex optimisation problem in the form
$$\begin{align}
f_0(x_1, \ldots, x_n) &\to \min \\
f_i(x_1, \ldots, x_n) & \leq 0, \quad i = 1, \ldots, m
\end{align}$$
where $f_0, f_1, \...
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Bounding a Solution of an SDP
It's common for convex optimization procedures to require a bounded region containing an optimal solution, either as input, like the initial ellipsoid of the ellipsoid method, or for run time bounds, ...
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Minimizing a convex piece-wise linear function of short $(\max, +)$ circuit length
If $a_{ij}$ is an $m \times n$ matrix of real numbers, and $b_j$ are $n$ more real numbers, then
$$\max_i \sum_j (a_{ij} x_j + b_j) \qquad (\ast)$$
is a convex piecewise linear function of $(x_1, \...
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On complexity of linear programming with quadratic equality/inequality constraints?
Feasibility test in Linear programming is in $P$ and in convex quadratic programming is in $P$. What is the maximum $k$ such that $n$-variable $m=poly(n)$ linear constraint feasibility test with $k$ ...
- 13.3k
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Has Khachiyan/Porkolob's convex integer optimization been implemented?
Khachiyan and Porkolab in 'Integer optimization on convex semialgebraic sets' gave an $O(ld^{ O(k^4)})$ algorithm to minimize a degree $d$ form with integer coefficients of binary length at most $l$ ...
- 13.3k
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What is wrong with this procedure to convert quadratic programming to convex quadratic programming?
Consider the feasibility quadratic program with constraint
$$\sum_{i=1}^nc_{i1}x_{i}\leq \ell_1$$
$$\vdots$$
$$\sum_{i=1}^nc_{it}x_{i}\leq \ell_t$$
$$\sum_{i,j=1}^na_{ij}x_{i}x_{j}+\sum_{i=1}^nb_{i}x_{...
- 13.3k
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Average case or beyond worse case analysis for non-convex optimization procedures?
I'm not sure if this is a well-formed question or not, but I thought I would ask to see if anyone is aware of related literature.
It is known that global optimization of non-convex functions is NP-...
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Looking for an easy/pedantic exposition of Renegar's famous result on polynomial optimization
In September $1989$, Renegar had this famous sequence of 3 papers titled, "On the Computational Complexity and Geometry of the First-order Theory of the Reals, Part I/II/III". I was wondering if ...
- 1,453
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How does one know what is not in a certain class of pseudo-distributions?
We consider working in the function space $\mathbb{R}^{\{ -1,1\}^n}$ where the expectation inner-product makes the juntas form a $2^n$ dimensional orthonormal basis.
Now say one has found a degree $...
- 1,453
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Properties of convex polytope of 0-1 matrices
Problem setting
Consider a set $ S = \big\{ 1,2,\cdots,n \big\}$. Now consider $k$ equal-sized subsets $S_i \subset S$ s.t of size $\big|S_i\big|=n' \;\forall i$.
Consider a $k\times k$ matrix $M$ ...
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Has compressed sensing been generalized to convex optimization problems?
Has the theory of "compressed sensing" been generalized to any classes of convex optimization problems? I need to analyze a problem of the type
$$\min ||x||_0, ~~~~ \mbox{ subject to } ~~~g(x) \leq 0$$...
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When is the duality gap of semidefinite programming (SDP) zero?
I haven't been able to find in the literature a precise characterization of the vanishing of the SDP duality gap. Or, when does "strong duality" hold?
For example, when one goes back and forth ...
- 1,453
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Do nested convex bodies have increasing "Volume/Surface Area" ratios? [closed]
Suppose we have two convex bodies $A$ and $B$, where $A \subseteq B$. Is it always true that $\mathrm{Vol}(A)/\mathrm{SurfaceArea}(A) \leq \mathrm{Vol}(B)/\mathrm{SurfaceArea}(B)$?
It's true in all ...
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Brute force search algorithm for semidefinite programming (representation of spectrahedron)
I was wondering if there exists a brute force search algorithm for semidefinite programming problems. Specifically, can we find finite number of points in the positive semidefinite cone such that for ...
- 449
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161
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Non-Linear Programming with \min operator in the constraint
Can the following non-linear program be solved in polynomial time? $c_{ij}$'s are constants and known. Each $c_{ij}$ is either -1 or 1.
\begin{align}
\text{maximize } &\sum_{i,j=1}^{m,n} c_{ij}...
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1
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The Average-case Complexity of Simplex Algorithm
I was wondering if there are any results on the average case complexity of the simplex algorithm. Let $A \in \mathbb{R}^{m \times n}$ be the matrix in the linear constraint. I know that Smale did ...
- 449
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First-order methods for solving SDP with geometric convergence or better
Is there any first-order method that can solve general SDP in a geometric (linear) rate? or super-geometric (super-linear) rate?
- 862
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When can a convex function induce submodularity?
Say I have a real valued convex function $f$ on the hypercube $[-1,1]^n$. Let $f'$ be the induced function on the discrete hypercube $\{-1,1\}^n$. Now I want to find a vertex on $\{-1,1\}^n$ on which ...
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Convergence of online convex optimization methods
I am new to this subject so this question might seem a bit trivial
Assume that in each round $t\in{{1,...T}}$ we choose $x_t\in K $ where $K$ is a compact and convex set, The common methods for ...
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Assignment of values for a set
Consider the following problem:
Input: the vertices of two $n$ dimensional axis-parallel cubes:
$\times_{i=1}^{n} [a_i,b_i] \subseteq [0,1]^n$ and
$\times_{i=1}^{n} [l_i,u_i] \subseteq [0,1]^n$.
...
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