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Questions tagged [optimization]

general questions about selecting a best element from some set of available alternatives.

11
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4answers
350 views

Reordering data to optimize for compression?

Are there any algorithms for reordering data to optimize for compression? I understand this is specific to the data and the compression algorithm, but is there a word for this topic? Where can I ...
1
vote
0answers
31 views

Problems rephrased as quadratic unconstrained binary optimization

I was impressed when i came across Quadratic unconstrained binary optimization (QUBO) recently, and saw how one can rephrase many combinatorial problems into questions about optima of binary functions....
2
votes
1answer
100 views

Intuitive explanation behind Goemans-Williamson randomized rounding

A very simple randomized cut algorithm achieves $1/2$ of the optimal value: just choose each vertex to be in the cut with probability $1/2$, independently. Goemans-Williamson does something more ...
-1
votes
0answers
54 views

greedy performance

I have a set function $f:2^V\rightarrow R_+ $which is non negative monotone supermodular function with a property that $f(\{x\})$ is same for all $x\in V$($f(\{x_1\})=f(\{x_2\})=\dots =f(\{x_i\}),\...
1
vote
1answer
71 views

3 dimensional matching shortest solution NP-hard?

We have array of arbitrary number of elements - 3d vectors with positive integers components - for example ...
2
votes
1answer
351 views

Follow the Perturbed Leader for nonlinear cost functions

The famous FTPL algorithm [1] is analyzing linear cost function. Is there any generalized proof for nonlinear functions known? Note that in the last paragraph of [1] it says "It would be great to ...
2
votes
1answer
116 views

Generalizations of linear programming

Linear problems can be solved in polynomial time. So can semidefinite programs and, presumably, many other useful classes of optimization programs. Is there a survey/lecture notes describing ...
4
votes
1answer
377 views

Minimum Union-Sum Cost Path

I have a minimum cost path selection problem that is different from the usual shortest path in that each type of cost is accounted only once in the total cost of the path if multiple edges on the path ...
0
votes
2answers
70 views

Bellman-Ford with Non-edge-decomposable Path Weights

Consider a directed graph $G(V,E)$ with non-negative edge weights. Also, let us define the weight of a path as non-edge-decomposable, that is, the weight of a path cannot be written as the sum of a ...
1
vote
0answers
121 views

Star seperators to explain computational complexity of algorithms on a class of graphs?

A lot of NP-hard optimization problems on graphs which are perfect become solvable in polynomial time. Unfortunately, the class of graphs that arise in my problem are not perfect. The graphs can be ...
2
votes
1answer
90 views

Maximum-minimum satisfiability

In MAX-SAT, given a formula, we want to maximize the number of satisfied clauses: given a formula $\phi = c_1 \cap \cdots \cap c_n$, where each $c_i$ is a disjunction, we want to find the largest $k\...
3
votes
2answers
108 views

Minimum relevant variables in linear system - additive approximation

In the problem Minimum Relevant Variables in Linear System (Min-RVLS), the input is a linear system, e.g.: $$ A x = b $$ and the goal is to find a solution $x$ with as few nonzero variables as ...
1
vote
1answer
91 views

Optimal evaluation of polynomials / rational functions

A common way to compute the value a polynomial is to write it in Horner form. However, this isn't always the fastest way to evaluate it. Setting aside concerns of numerical precision, take the ...
4
votes
1answer
561 views

Knapsack with dependent profits (pairs of items)

I'm working on a problem which MAY be reduced to the following version of Knapsack: Suppose two items $e_i$ and $e_j$ have profit $p_i$ and $p_j$ respectively. However, if both items are present in ...
1
vote
0answers
30 views

Pass ordering for greedy local search algorithms

Apologies in advance for the slightly general question - I'm really looking for pointers to research / good keywords to look for. I have a problem with the following setup: I have a (finite) totally ...
1
vote
1answer
129 views

Is this a knapsack problem?

I have a set of $K$ keywords. Each of this keywords can have a set of bids from $1\$,\ldots, N\$$. For each bid for a keyword, it will get a specific amount of clicks and a specific cost. Clicks and ...
1
vote
0answers
36 views

Back-propagation for computing derivative of certain line integral

Consider a function F (think of neural networks) with two sets of parameters: (1) model parameters $\mathbf{w}$, and (2) input data ${\bf x} \in {\mathbb R}^d$. Fix $i \in [d]$, consider the following ...
1
vote
0answers
35 views

Best approach for allocation problem

I am a bit rusty on optimization algorithms and need an advice. This is my problem: I have n images (with width and ...
2
votes
1answer
95 views

Stochastic gradient methods and risk of neural nets

Under many situations it is currently provable that we can minimize the risk of neural nets using stochastic gradient based algorithms. For example : https://arxiv.org/abs/1811.03804, https://arxiv....
3
votes
1answer
844 views

Minimal Cost of Eulerian Path

Problem: Given a planar (undirected and mostly sparse) graph with an Eulerian Path, we introduce a cost function f: (v, e1, e2) for all two edges e1 and e2 that share a vertex v. The function also ...
2
votes
1answer
99 views

Minimization version of matrix p-norms?

I considered a minimization version of matrix p-norms, defined for a matrix $A$ by $$ f_p(A)= \min_{x\neq 0} \frac{||Ax||_p}{||x||_p}. $$ Notice that $f_p(A) = 0$ if and only if $A$'s columns are ...
1
vote
1answer
44 views

Problem property name where an optimal solution in a graph can be used as a solution in any subgraph

Suppose one is given a graph optimization problem where the optimal solution $S$ for the problem on graph $G$ can be used as a solution for any subgraph of $G$. In other words, given $S$ is an optimal ...
9
votes
0answers
113 views

Shortest string in the intersection of regular languages

Inspired by https://codegolf.stackexchange.com/questions/53310/shortest-universal-maze-exit-string Each of the 138,172 valid mazes can be represented as a DFA with 9 states (including starting and ...
3
votes
0answers
69 views

Linear optimization over intersection of totally unimodular matrices

I am currently dealing with a problem of the following form \begin{alignat}{2} &\underset{x, y \in \mathbb{R}^n}{{\text{min}}} && e^T x \nonumber\\ &\text{sub to} \hspace{0.05in}&&...
6
votes
0answers
140 views

Optimal set union tree

Suppose we have a ground set of $n$ elements and $m$ sets are defined over them $S_i \subseteq [n]$. Think of the following procedure: At each step take two of the sets, take the union, and add the ...
10
votes
2answers
679 views

Set optimization problem - is it np-complete?

Set $S=\{e_1,\cdots,e_n\}$ is given. For each element $e_i$, we have weight $w_i>0$ and cost $c_i>0$. The goal is findIng the subset $M$ of size $k$ that maximize the following objective ...
3
votes
1answer
217 views

Optimal partition according to partition cardinality

Given $N$ sets of integers $S_1, \ldots,S_N$ with $|S_i| \le K$. We want to partition those sets such that the union of all sets in any given partition doesn't contain more than $K$ elements. Can ...
1
vote
1answer
64 views

Reference request — minimizing a non-increasing submodular function with (upper bound) cardinality constraint

Suppose a set function $f(S)$ is submodular and non-increasing, meaning that for any $S'\subset S$, $f(S') \geq f(S)$. The problem is to minimize $f(S)$ s.t. $|S| \leq k$. I am wondering if there are ...
1
vote
1answer
389 views

Optimal value of a semidefinite program

Is a local optimum value of a SDP always the global one? If not, what are the conditions for that?
4
votes
0answers
529 views

Optimizing Maximum Weighted Matching (Edmonds Blossom)

Background: I've ported Edmonds Blossom Algorithm with Maximum Weighted Matching to Java: https://github.com/simlu/EdmondsBlossom/blob/master/src/Blossom.java The original Python implementation ...
1
vote
1answer
72 views

Minimum graph cut with constraints

Given an undirected acyclic raph $G = \{V,E\}$, with each edge $e$ having weight $c_e$in the range $[-\infty, +\infty] $, I want to compute a partition of the graph into $N$ disjoint sets $G_i, i=1,......
1
vote
2answers
82 views

Partitioning a square for optimal queries

I have a square plate of size 1x1, full of lots of skittles. I want to eat all of the skittles, but the only way I can get the skittles is through these two oracles: $f(x, y, r)$ tells me how many ...
1
vote
0answers
27 views

Variability of gradient estimates and convergence rate in stochastic gradient descent/ascent

I am aware that convergence in stochastic gradient problems is very sensitive to the variance of your gradient estimator. One issue I'm running into is that the gradient is a random vector and so ...
1
vote
0answers
55 views

Minimization of the maximal adjacent integer sums on a circle

Arrange $\{1,2,\cdots,n\}$ on a circle. What are the arrangements that minimize the maximal sum of all adjacent $k$ integers? For specific and low $n$'s it has been pointed out in math.stackexchange....
3
votes
1answer
47 views

Algorithm for finding smallest set and instanciation for a given constraint system

I have a system of constraints described by a set of clauses of the form $x_1 \neq x_2 \lor \dots \lor x_{i-1} \neq x_i$, for instance: ...
3
votes
0answers
47 views

Primal/Dual of the Lasserre/ SOS SDP hierarchy

Sum of Squares proofs and the Lasserre hierarchy can both be stated as SDPs. It is often claimed without proof that these SDPs are dual to each other, although I do not see that this is obvious. I was ...
4
votes
0answers
83 views

Showing hardness of maximizing stochastic objective function over graph

Consider a graph $G = (V, E)$ with $n$ vertices and $m$ edges. Each vertex $v_i$ can take positive value $a_i$ with probability $p_i$ and value $0$ with probability $1-p_i$. The challenge is to ...
0
votes
1answer
46 views

About using smoothness of the Hessian for getting to approximate criticality of a non-convex objective

Is there any algorithm which shows that under the assumption of Lipschitz smoothness of the Hessian of a non-convex function one can get to its critical point faster?
1
vote
0answers
32 views

Getting to local/global minima with (stochastic) gradient descent on non-convex objectives

Is there any class of non-convex objective functions for which (stochastic) gradient descent can provably get to a local or a global minima? (..maybe in the approximate sense like a point such that ...
1
vote
1answer
122 views

Monotone supermodular function minimization under partition matroid constraints

Is there a known approximation algorithm for the problem of minimizing a monotone (non increasing) supermodular function under partition matroid constraints ?
0
votes
1answer
39 views

How do minimally violated k-mod cuts work (intuitive explanation)?

As a background, I am not a specialist in theoretical computer science. But I have to take an exam with research-level optimization topics, and I have to learn it on my own, without lectures or tutors....
2
votes
1answer
372 views

Maximum difference between two shortest paths

My problem is the following maximization problem: Given: A graph $G=(V,E)$, lower bounds $l \in \{0,1,..,K\}^E$ and upper bound $u \in \{0,1,..,K\}^E$ for the edge weights, a source $s$ and two ...
2
votes
0answers
165 views

Is Non-linear Constrained Optimal Exact Cover NP-Hard?

Playing around I ran into a problem which looks like a Exact Set Covering / Partition Problem, but I am unable to find a reduction to categorize the complexity of the problem, despite it looks NP-Hard....
3
votes
2answers
282 views

NP-hardness of finding 0-1 vector to maximize rows of {-1, +1} matrix

Consider the following discrete optimization problem: given a collection of $m$-dimensional vectors $\{ v_1, \dots, v_n \}$ with entries in $\{-1, +1\}$, find an $m$-dimensional vector $x$ with ...
3
votes
1answer
5k views

Difference between weak duality and strong duality?

For an optimization problem $(P)$ and its dual $(D)$, I have read about two concepts: Weak Duality, and strong Duality. What I don't understand is how they are different: Weak duality: If $\bar{x}$ ...
4
votes
1answer
465 views

Understanding the No Free Lunch Theorem

I came across the No Free Lunch Theorem via Jürgen Schmidhuber's paper on Universal Search and there were a couple remarks on NFL which stood out to me. The first was that we can't define a uniform ...
3
votes
0answers
119 views

Stacks serving interval storage requests

An interval storage request is represented by a tuple $(s,t,v)$ satisfying $s<t$, meaning that the value $v$ needs to be stored from time $s$ to time $t$. A stack serves the request $(s,t,v)$ in ...
3
votes
0answers
65 views

Minimizing a sum of thresholded quadratics

Let $W_1, \ldots, W_k$ be positive semi-definite matrices, $b_1, \ldots, b_k$ be vectors, and $a_1, \ldots a_k$, $c_1, \ldots, c_k$ be scalars. How difficult is it to find an approximate minimum of ...
5
votes
0answers
170 views

Which well-known algorithmic problem is this an instance of?

Consider that you have n counters initialised with numbers $M_1 \dots M_n$. In each round you decrement exactly $k$ out of these counters. Keep doing this until at least $n-k+1$ counters are zero, so ...
1
vote
0answers
110 views

QUBO formulation of a discrete-variable optimization problem

I am facing a non-linear, discrete optimization problem, which I can formulate in this abstract manner: I have a certain non-analytic non-linear real-valued function $f:S \to \mathbb{R}$ which takes ...