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

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2
votes
0answers
87 views

Optimal additive basis for decomposing/partitioning an integer as a sum of two integers

I'm going to be given a positive integer $z$, and I want to find an optimal basis $B$ that is good for $z$. A basis $B$ is a multiset of positive integers. The basis $B$ is considered good for $z$ ...
11
votes
0answers
159 views

the largest element of a matrix product

Given two matrices, I'm interested in finding the largest element of their product. I wonder if it's possible to do it significantly faster than the matrix multiplication the solution seems to ...
1
vote
1answer
146 views

Ising spin vs Pauli spin matrices [closed]

Are Ising spins scalar or operators? I am not a condensed matter physicist hence having some confusion. I have learnt about Ising models from adiabatic quantum algorithm papers. For example this ...
3
votes
3answers
533 views

Maximize quadratic function subject to linear constraints

Can one maximize $\sum_i c_i x_i^2$ where the $c_i$ are constants (possibly negative), subject to linear constraints over the $x_i$? This paper seems to come close to answering "no." They show it is ...
0
votes
0answers
2k views

Quantum annealing vs adiabatic quantum computation

I had this impression that quantum annealing is an optimization technique which may or may not produce exact solutions. On the other hand adiabatic quantum computation always gives exact solutions ...
3
votes
0answers
64 views

Closest Vector Problem with sparse basis and target vector

The Closest Vector Problem (and related problems) is random self-reducible and in general is NP-Hard, making it a useful tool in cryptography research and post-quantum public key crypto. For a variety ...
9
votes
0answers
160 views

Expected length of a self-avoiding random walk

We're given an arbitrary graph $G=(V,E)$. A self-avoiding random walk is a random walk such that in each step, a random neighbour which was not previously visited is chosen, if such one exists. ...
1
vote
0answers
256 views

Confusion related to L1 and L2 svm [closed]

I have this confusion related to L1 and L2 svm. It is given in this paper that The dual problem is given by $$ min(\alpha) = 1/2*\alpha^T\hat Q\alpha - e^T\alpha $$ subject to $$ 0 <= \alpha_i ...
1
vote
0answers
92 views

Poly-time Algorithm for Non-Linear Optimization

As we know, linear programming is one of the most basic area of optimization theory, and computing an optimal solution can be excuted within poly-time. My question is about an extention of this ...
2
votes
2answers
249 views

Undecidability of program optimization

A program is an encoded Turing Machine. And a size optimizer of a program is a TM $M_1$ such that: On any input $M$, $M_1$ outputs $M_{min}$ such that $M_{min}$ is the shortest TM which is ...
3
votes
1answer
186 views

A weighted sorting problem

Given a data matrix $D=[d_1 ... d_N]$, one would like to sort it in terms of rows such that the weighted distance of sorted $d$s to a target vector $y$ is being minimized. It can be formulated as ...
1
vote
0answers
134 views

What does “no integrality gap” implies?

I'm currently working on a linear time heuristic for the rectangle decomposition of binary matrix. This problem has a polynomial time solution, which in our case it too slow for large scale ...
-2
votes
2answers
220 views

how do you turn an algorithm for a decision problem into an algorithm for an optimization problem?

It is well-known, I believe, that theoretically, in quite a few cases, an algorithm that solves a decision problem can be turned into an algorithm that solves the corresponding optimization problem. ...
2
votes
0answers
68 views

Small area containing large amount of patterns

The problem: I come across a theoretical problem when designing characters for electron-Beam lithography. Abstractly, given an integer $m$, let $\mathcal{M}$ be the set of $(0,1)$-matrix $A_{p\times ...
3
votes
0answers
190 views

Slightly Faster Exponential Algorithm for Integer Programming with Multi-linear Variables

Integer programing is one of the most narutal optimization tools. As an analogy of DNF or CNF in the Boolean function theory, we can consider the following equation. $x_{1}x_{2}x_{3}+$ ...
1
vote
0answers
143 views

What is the shape of cost function of weighted graph matching problem?

According to Umeyama, the weighted graph matching problem can be formulated as $min_P || PA_GP^T - A_H ||$ s.t. $P$ is a permutation matrix. where $A_G$ and $A_H$ are n-by-n matrices If we relax ...
3
votes
0answers
119 views

Existence of complete “asymptotic” optimization problems

Define a function $u:\lbrace 0,1 \rbrace^* \rightarrow \mathbb{R}$ to be an asymptotic optimization problem when the following conditions hold: There is an algorithm $U$ which computes the first $n$ ...
8
votes
2answers
440 views

Generating interesting combinatorial optimization problems

I'm teaching a course on meta-heuristics and need to generate interesting instances of classic combinatorial problems for the term project. Let's focus on TSP. We are tackling graphs of dimension ...
0
votes
0answers
78 views

Nonmetric TSP and Functional Compleixty Classes

Non-metric TSP that is TSP and some instance is not hold the triangle inequality is NP-hard by gap-reduction method. Is this general TSP a complete problem in some functional complexity classes ? ...
-1
votes
1answer
278 views

Integer compression

I'm looking to devise a scheme for compressing integers which have a known sampled distribution (they might be clustered around a value, say, or have several areas of differing density). So far, I've ...
2
votes
0answers
93 views

Benchmarks for approximation algorithms

I'm working on a Haskell library for approximation algorithms. In particular, I'm working on Partition, Knapsack, Vertex Cover, and possibly a few others. Of course, I'd like to benchmark my library ...
1
vote
0answers
168 views

Properties of the subgradient method

The subgradient algorithm for minimizing a convex function $f(x)$ is the update rule $$ x(t+1) = x(t) - \alpha(t) d(t)$$ where $d(t)$ is any subgradient of $f(x)$ at $x(t)$ and $\alpha(t)$ is a ...
4
votes
1answer
137 views

Splitting line segments with a line

Given is a finite set $S$ of line segments in the plane. I am interested in finding a line $l$ which splits some segments in $S$ into two, thus yielding a new set of line segments $S'$. Here ...
5
votes
1answer
234 views

Effective algorithm of searching the “nearest” doubly stochastic matrix

Given a data matrix $D$, is there any effective algorithm to solve the optimization problem $\min_Q || D - Q ||_F$ such that $Qe=e$, $e^TQ=e^T$, and $Q_{i,j} \geq 0 $ $\forall i,j$, where ...
12
votes
1answer
274 views

Numerical stability of Simplex method

The simplex algorithm is often treated either within real arithmetic, or in the discrete world with exact computations. However, it seems to be implemented most often with floating-point arithmetic. ...
2
votes
1answer
57 views

Covering only one of two types of objects in a cartesian space using minimum number of rectangles

There is a side problem in my research that I believe should be a known problem. I do not want to spend lots of time on a problem that already has been studied, but I do not have a name for the ...
2
votes
1answer
138 views

NP-complete problems related to Minimizing Variance

I am interested in references to NP-complete problems that involve some non-linear terms (e.g. quadratic terms). So far I am aware of the "Quadratic Assignment problem" and "Quadratic Programming". ...
9
votes
1answer
539 views

Minimum spanning tree over all vertex matchings

I ran into this matching problem for which I am unable to write down a polynomial time algorithm. Let $P, Q$ be complete weighted graphs with vertex sets $P_V$ and $Q_V$, respectively, where $|P_V| ...
4
votes
1answer
136 views

Nearly-Eulerian Tours

The Eulerian Tour problem is of course a well-studied classical problem in graph theory (Wikipedia article). This question concerns non-Eulerian graphs; i.e., graphs that contain one or more ...
1
vote
0answers
64 views

Can I apply the constraint while constructing the Lagrangian?

Consider the problem: $\min_X ||XAX^T||_F$ s.t. $X^TX=I$ If A and X are real matrices, the lagrangian will be $tr(XAX^TXA^TX^T)+\sum_i\alpha_i(x_i^Tx_i-1)+\sum_i\sum_{j\neq i}\beta_{ij} q_i^Tq_j$ ...
3
votes
1answer
260 views

Stochastic version of a strongly NP-Complete problem

Does a strongly NP-Complete problem remain strongly NP-Complete if the variable set on which the objective/cost function depends are made stochastic ? The problem Tree CVRP(Capacitated Vehicle ...
7
votes
1answer
298 views

NP-hardness of an optimization problem

While studying a problem in algorithmic game theory I got interested in the complexity of the following optimization question: Problem Given: ground set $U = [n] = \{1,\ldots,n\}$ given by $n$, ...
13
votes
1answer
602 views

Exact algorithms for non-convex quadratic programming

This question is about quadratic programming problems with box constraints (box-QP), i.e., optimisation problems of the form minimise $f(\mathbf{x}) = \mathbf{x}^T A \mathbf{x} + \mathbf{c}^T ...
14
votes
2answers
867 views

0-1 Linear Programming: computing the Optimal Formulation

Consider the $n$ dimensional space $\{0,1\}^n$, and let $c$ be a linear constraint of the form $a_1x_1 + a_2x_2 + a_3x_3 +\ ...\ + a_{n-1}x_{n-1} + a_nx_n \geq k$, where $a_i \in \mathbb{R}$, $x_i \in ...
6
votes
0answers
394 views

Which problems in graph theory can be stated as quadratic programs?

There seem to be many very interesting problems in graph theory that can be written in the form of maximizing/minimizing a quadratic form on either the Adjacency ${\bf A}$ or the Laplacian matrix ...
5
votes
1answer
242 views

Splitting a graph into minimum number of subpaths

I have an ordinary weighted graph. I need to traverse every edge in the graph at least once. BUT I must do it in subpaths of maximum length L. Those subpaths need not be connected to each other. There ...
1
vote
0answers
132 views

Trace minimization with an orthogonality constraint

For positive semi-definite matrices $A,B$, how can I find an $X$ that minimizes $\text{Trace}(AX^TBX$) under the constraint: that $X$ is orthogonal. All the matrices have real entries and $A,B$ are ...
2
votes
1answer
944 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 $\bar{x}$ ...
2
votes
1answer
842 views

Is quantum annealing faster than simulated annealing/genetic/other state-of-the-art optimization algorithms?

There's the idea of quantum annealing being used to solve optimization problems in terms of a QUBO problem for D-Wave's quantum algorithm. I understand that the advantage of quantum annealing as ...
2
votes
1answer
783 views

What are some good references for mathematical optimization for the layman?

I've been getting myself involved with this topic and would like to read more to gain a conceptual understanding of the various techniques and what each one is trying to achieve and their 'idea' ...
8
votes
2answers
668 views

How/Why are linear systems so crucial to computer science?

I've begun to get involved with Mathematical Optimization quite recently and am loving it. It seems a lot of optimization problems can be easily expressed and solved as linear programs (e.g. network ...
3
votes
1answer
296 views

Is this multiprocessor scheduling problem with overlaps NP-Hard?

The problem statement is: "Given a set $J$ of jobs where job $J_i$ has length $L_i$ and a number of processors $m$, jobs have inter-overlapping (For example, if job $J_i$ and $J_k$ are assigned to the ...
8
votes
1answer
227 views

Maximizing a submodular function of two sets with different size constraints

I have two totally distinct domains (apples and oranges) and I have a function $f$ that takes a set of objects from the first domain and a set of objects from the second domain and returns a real ...
2
votes
0answers
199 views

Maximizing a convex function where the objective function is separable but the search space is not

The problem statement is Given convex functions $f_i$ over $X$, find $$\arg\max_{x\in X} \sum_i f_i(x)$$ Does this kind of problem structure allow one to use specific strategies to solve the ...
1
vote
1answer
582 views

Vehicle routing problem over Manhattan distances

I am looking for references to the variant of the vehicle routing problem over Manhattan distance metric where the aim is to optimize the number of tours starting at the depot. Is the following ...
5
votes
1answer
810 views

Implementation that solves minimum set cover

Does anyone know of any tools that solve the approximate minimum set cover problem? I know of the greedy algorithm (which is straightforward to implement myself), but I've also been reading about ...
1
vote
0answers
235 views

Optimizing along a cube $s=\{0,1\}^n$

I am doing an optimization on a n-dimensional cube. That means that every solution is a set of $0$ and $1$, hence $s=\{0,1\}^n$. Most optimization algorithms though need a differential to work. E.g. ...
4
votes
0answers
690 views

Can the Hungarian method be used with real edge weights?

I had a problem where I need to apply bipartite weighted matching on a graph where the edge weights are real (positive and negative). I have looked at several implementations of the Hungarian method ...
0
votes
1answer
336 views

To what extent is it possible to use genetic algorithms to make wind mill turbine blades more efficient?

I recently watched this video on youtube. It featured someone explaining how he used genetic algorithms to improve the efficiency of wind mill turbines by finding the optimal shape for the blades. ...
5
votes
0answers
180 views

Maximizing difference of a submodular and a modular function

I'm considering a network planning problem which is stated as follows: From the given ground set $\mathcal{V}$, select $\mathcal{A} \subseteq \mathcal{V}$ such that \begin{equation} f(\mathcal{A}) - ...