scholarly journals New Applied Problems in the Theory of Acyclic Digraphs

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 45
Author(s):  
Gurami Tsitsiashvili ◽  
Victor Bulgakov
Keyword(s):  
Optimization Problems ◽  
Maximum Flow ◽  
Acyclic Digraph ◽  
Strongly Connected ◽  
Minimum Number

The following two optimization problems on acyclic digraph analysis are solved. The first of them consists of determining the minimum (in terms of volume) set of arcs, the removal of which from an acyclic digraph breaks all paths passing through a subset of its vertices. The second problem is to determine the smallest set of arcs, the introduction of which into an acyclic digraph turns it into a strongly connected one. The first problem was solved by reduction to the problem of the maximum flow and the minimum section. The second challenge was solved by calculating the minimum number of input arcs and determining the smallest set of input arcs in terms of the minimum arc coverage of an acyclic digraph. The solution of these problems extends to an arbitrary digraph by isolating the components of cyclic equivalence in it and the arcs between them.

scholarly journals An Intrinsically-Motivated Approach for Learning Highly Exploring and Fast Mixing Policies

2020 ◽  
Vol 34 (04) ◽  
pp. 5232-5239
Author(s):  
Mirco Mutti ◽  
Marcello Restelli
Keyword(s):  
Optimal Policy ◽  
Optimization Problems ◽  
Learning Algorithm ◽  
Empirical Evaluation ◽  
Steady State Distribution ◽  
State Distribution ◽  
State Action ◽  
Uniform State ◽  
Minimum Number ◽  
Efficient Learning

What is a good exploration strategy for an agent that interacts with an environment in the absence of external rewards? Ideally, we would like to get a policy driving towards a uniform state-action visitation (highly exploring) in a minimum number of steps (fast mixing), in order to ease efficient learning of any goal-conditioned policy later on. Unfortunately, it is remarkably arduous to directly learn an optimal policy of this nature. In this paper, we propose a novel surrogate objective for learning highly exploring and fast mixing policies, which focuses on maximizing a lower bound to the entropy of the steady-state distribution induced by the policy. In particular, we introduce three novel lower bounds, that lead to as many optimization problems, that tradeoff the theoretical guarantees with computational complexity. Then, we present a model-based reinforcement learning algorithm, IDE3AL, to learn an optimal policy according to the introduced objective. Finally, we provide an empirical evaluation of this algorithm on a set of hard-exploration tasks.

scholarly journals Integer Programming on the Junction Tree Polytope for Influence Diagrams

2020 ◽  
Vol 2 (3) ◽  
pp. 209-228
Author(s):  
Axel Parmentier ◽  
Victor Cohen ◽  
Vincent Leclère ◽  
Guillaume Obozinski ◽  
Joseph Salmon
Keyword(s):  
Graphical Model ◽  
Optimization Problems ◽  
Valid Inequalities ◽  
Random Variables ◽  
Mixed Integer ◽  
Influence Diagrams ◽  
Probabilistic Graphical Model ◽  
Computationally Efficient ◽  
Junction Tree ◽  
Acyclic Digraph

Influence diagrams (ID) and limited memory influence diagrams (LIMID) are flexible tools to represent discrete stochastic optimization problems, with the Markov decision process (MDP) and partially observable MDP as standard examples. More precisely, given random variables considered as vertices of an acyclic digraph, a probabilistic graphical model defines a joint distribution via the conditional distributions of vertices given their parents. In an ID, the random variables are represented by a probabilistic graphical model whose vertices are partitioned into three types: chance, decision, and utility vertices. The user chooses the distribution of the decision vertices conditionally to their parents in order to maximize the expected utility. Leveraging the notion of rooted junction tree, we present a mixed integer linear formulation for solving an ID, as well as valid inequalities, which lead to a computationally efficient algorithm. We also show that the linear relaxation yields an optimal integer solution for instances that can be solved by the “single policy update,” the default algorithm for addressing IDs.

Algorithms for Convex Optimization

2021 ◽  
Author(s):  
Nisheeth K. Vishnoi
Keyword(s):  
Convex Optimization ◽  
Data Science ◽  
Optimization Problems ◽  
Algorithm Design ◽  
Continuous Optimization ◽  
Maximum Flow ◽  
Maximum Matching ◽  
Science Data ◽  
Mirror Descent ◽  
Continuous Optimization Problems

In the last few years, Algorithms for Convex Optimization have revolutionized algorithm design, both for discrete and continuous optimization problems. For problems like maximum flow, maximum matching, and submodular function minimization, the fastest algorithms involve essential methods such as gradient descent, mirror descent, interior point methods, and ellipsoid methods. The goal of this self-contained book is to enable researchers and professionals in computer science, data science, and machine learning to gain an in-depth understanding of these algorithms. The text emphasizes how to derive key algorithms for convex optimization from first principles and how to establish precise running time bounds. This modern text explains the success of these algorithms in problems of discrete optimization, as well as how these methods have significantly pushed the state of the art of convex optimization itself.

GENETIC ALGORITHMS AND REINFORCEMENT LEARNING FOR THE TACTICAL FIXED INTERVAL SCHEDULING PROBLEM

2001 ◽  
Vol 10 (01n02) ◽  
pp. 23-38 ◽  
Author(s):  
EUGENE SANTOS ◽  
XIAOMIN ZHONG
Keyword(s):  
Reinforcement Learning ◽  
Optimization Problems ◽  
Structural Characteristics ◽  
Fixed Interval ◽  
Learning System ◽  
Scheduling Problem ◽  
Interval Scheduling ◽  
Minimum Number ◽  
Identical Machine ◽  
Fixed Interval Scheduling

Discrete optimization problems are usually NP hard. The structural characteristics of these problems significantly dictate the solution landscape. In this paper, we explore a structure-based approach to solving these kinds of problems. We use a reinforcement learning system to adaptively learn the structural characteristics of the problem, hereby decomposing the problem into several subproblems. Based on these structural characteristics, we develop a Genetic Algorithm by using structural operations to recombine these subproblems together to solve the problem. The reinforcement learning system directs the GA. We test our algorithm on the Tactical Fixed Interval Scheduling Problem(TFISP) which is the problem of determining the minimum number of parallel non-identical machine such that a feasible schedule exists for a given set of jobs. This work continues our work in exploiting structure for optimization.

A UNIFYING FRAMEWORK FOR GENERALIZED CONSTRAINT ACQUISITION

2008 ◽  
Vol 17 (05) ◽  
pp. 803-833 ◽  
Author(s):  
XUAN-HA VU ◽  
BARRY O'SULLIVAN
Keyword(s):  
Optimization Problems ◽  
Constraint Satisfaction Problem ◽  
Active Role ◽  
Probabilistic Constraints ◽  
Practical Applications ◽  
Minimum Number ◽  
Programming Techniques ◽  
The Difference ◽  
Boolean Optimization ◽  
Constraint Acquisition

When a practical problem can be modeled as a constraint satisfaction problem (CSP), which is a set of constraints that need to be satisfied, it can be solved using many constraint programming techniques. In many practical applications, while users can recognize examples of where a CSP should be satisfied or violated, they cannot articulate the specification of the CSP itself. In these situations, it can be helpful if the computer can take an active role in learning the CSP from examples of its solutions and non-solutions. This is called constraint acquisition. This paper introduces a framework for constraint acquisition in which one can uniformly define and formulate constraint acquisition problems of different types as optimization problems. The difference between constraint acquisition problems within the framework is not only in the type of constraints that need to be acquired but also in the learning objective. The generic framework can be instantiated to obtain specific formulations for acquiring classical, fuzzy, weighted or probabilistic constraints. The paper shows as an example how recent techniques for acquiring classical constraints can be directly obtained from the framework. Specifically, the formulation obtained from the framework to acquire classical CSPs with the minimum number of violated examples is equivalent to a simple pseudo-boolean optimization problem, thus being efficiently solvable by using many available optimization tools. The paper also reports empirical results on constraint acquisition methods to show the utility of the framework.

scholarly journals Weapon Target Assignment

2021 ◽  
Author(s):  
Mohammad Babul Hasan ◽  
Yaindrila Barua
Keyword(s):  
Integer Programming ◽  
Objective Function ◽  
Optimization Problems ◽  
Optimization Techniques ◽  
Nonlinear Integer Programming ◽  
Integer Programming Problem ◽  
Combinatorial Optimization Problems ◽  
Target Assignment ◽  
Minimum Number ◽  
Np Complete

This chapter is mainly based on an important sector of operation research-weapon’s assignment (WTA) problem which is a well-known application of optimization techniques. While we discuss about WTA, we need some common terms to be discussed first. In this section, we first introduce WTA problem and then we present some prerequisites such as optimization model, its classification, LP, NLP, SP and their classifications, and applications of SP. We also discuss some relevant software tools we use to optimize the problems. The weapon target assignment problem (WTA) is a class of combinatorial optimization problems present in the fields of optimization and operations research. It consists of finding an optimal assignment of a set of weapons of various types to a set of targets in order to maximize the total expected damage done to the opponent. The WTA problem can be formulated as a nonlinear integer programming problem and is known to be NP-complete. There are constraints on weapons available of various types and on the minimum number of weapons by type to be assigned to various targets. The constraints are linear, and the objective function is nonlinear. The objective function is formulated in terms of probability of damage of various targets weighted by their military value.

scholarly journals Extension of Gyárfás-Sumner Conjecture to Digraphs

10.37236/9906 ◽  
2021 ◽  
Vol 28 (2) ◽  
Author(s):  
Pierre Aboulker ◽  
Pierre Charbit ◽  
Reza Naserasr
Keyword(s):  
Chromatic Number ◽  
Directed Path ◽  
Complete Multipartite Graph ◽  
Color Class ◽  
Acyclic Digraph ◽  
Minimum Number ◽  
Multipartite Graph ◽  
Complete Multipartite ◽  
Dichromatic Number

The dichromatic number of a digraph $D$ is the minimum number of colors needed to color its vertices  in such a way that each color class induces an acyclic digraph. As it generalizes the notion of the chromatic number of graphs, it has become the focus of numerous works. In this work we look at possible extensions of the Gyárfás-Sumner conjecture. In particular, we conjecture a simple characterization  of sets $\mathcal F$ of three digraphs such that every digraph with sufficiently large dichromatic number must contain a member of $\mathcal F$ as an induced subdigraph.  Among notable results, we prove that oriented $K_4$-free graphs without a directed path of length $3$ have bounded dichromatic number where a bound of $414$ is provided. We also show that an orientation of a complete multipartite graph with no directed triangle is $2$-colorable. To prove these results we introduce the notion of nice sets that might be of independent interest.

DESIGN OF CONTROLLABLE LEADER–FOLLOWER NETWORKS VIA MEMETIC ALGORITHMS

2021 ◽  
pp. 2150004
Author(s):  
SHAOPING XIAO ◽  
BAIKE SHE ◽  
SIDDHARTHA MEHTA ◽  
ZHEN KAN
Keyword(s):  
Genetic Algorithms ◽  
Optimal Design ◽  
Optimization Problems ◽  
Memetic Algorithms ◽  
Networked Systems ◽  
Efficiency And Effectiveness ◽  
Minimum Number ◽  
Simulation Results ◽  
Leader Selection ◽  
Network Controllability

In many engineered and natural networked systems, there has been great interest in leader selection and/or edge assignment during the optimal design of controllable networks. In this paper, we present our pioneering work in leader–follower network design via memetic algorithms, which focuses on minimizing the number of leaders or the amount of control energy while ensuring network controllability. We consider three problems in this paper: (1) selecting the minimum number of leaders in a pre-defined network with guaranteed network controllability; (2) selecting the leaders in a pre-defined network with the minimum control energy; and (3) assigning edges (interactions) between nodes to form a controllable leader–follower network with the minimum control energy. The proposed framework can be applied in designing signed, unsigned, directed, or undirected networks. It should be noted that this work is the first to apply memetic algorithms in the design of controllable networks. We chose memetic algorithms because they have been shown to be more efficient and more effective than the standard genetic algorithms in solving some optimization problems. Our simulation results provide an additional demonstration of their efficiency and effectiveness.

Multi-Objective Optimization Problems in Taguchi Parameter Design – A Literature Review

2015 ◽  
Vol 813-814 ◽  
pp. 1188-1192
Author(s):  
S. Rajarasalnath ◽  
K. Balasubramanian ◽  
N. Rajeswari
Keyword(s):  
Real Time ◽  
Manufacturing Process ◽  
Optimization Problems ◽  
Taguchi Methods ◽  
Quality Characteristic ◽  
Multi Objective Optimization ◽  
Multi Objective ◽  
Minimum Number ◽  
Set Up ◽  
Single Response

With the advent of latest technology, manufacturing process becomes so sophisticated and complicated that a single response variable (quality characteristic) can not reflect the true product quality and there is intense competition between market participants for cost and delivery (productivity) as well. Any manufacturing process in the present state requires multi objective optimization model to optimize quality, cost and productivity simultaneously. Optimum prediction is critical as it requires lot of experiments for data capturing which involves time and cost. In the present manufacturing set up, it is essential to identify optimum parametric combination for multi objective function real time problems with lesser experiments and lesser effort with better accuracy. Taguchi method is a well known fraction factorial design, which requires minimum number of trials for Identifying optimum parametric combination in real time problems.In this paper an attempt is made to review the literatures of various methods used by researchers for multi - objective optimization problems using Taguchi methods.

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