Skip to content
Free planNo credit card requiredStart writing
AI for Research home
AI for Research home
Dashboard
Sign inGet started

Search 250 million research papers

Read the abstract, see who wrote it and who cites it, and get the citation in APA, MLA, Chicago or Harvard.

About 8,520 papers for “local minimum problem”

Subgoal chaining and the local minimum problem

Jonathan P. Lewis, Michael Weir · 2003

It is well known that performing gradient descent method on fixed surfaces may result in poor travel through getting stuck in local minima and other surface features. Subgoal chaining in supervised learning is a method to improve travel for neural networks by directing local variation in the surface during training. This paper shows that linear subgoal chains such as those used in ERA are not sufficient to overcome the local minimum problem and examines nonlinear subgoal chains as a possible alternative.

Cited by 4

LOCAL SEARCH FOR THE MINIMUM FUNDAMENTAL CYCLE BASIS PROBLEM

E. Amaldi, L. Liberti, N. Maculan et al. · 2008

Let G=(V,E) be a bi-connected graph with non-negative weights on its m edges. Let n be the number of vertices of G. With respect to any spanning tree T of G we have a fundamental cycle basis of G formed by the m-n+1 fundamental cycles, each one corresponding to one of the co-tree edges. The problem of determining one such basis minimising the sum of the length of its cycles (MinFCB) is known to be NP-hard. Applications exist in organic chemistry, periodic scheduling, and electrical networks. A recent result proves that no polynomial time approximation scheme can exist for MinFCB unless P=NP. The problem of determining a tree with respect to which the longest fundamental cycle is minimum is also NP-hard. When fundamentality is not required, that is when one looks for a basis of the cycle space of the graph, both problems are solvable in polynomial time. Constructive heuristics have been proposed in the literature, which are fast and an obvious first choice for very large instances. However they tend to produce solutions which are far from optimal. We report about on-going research on Local Search approaches for MinFCB. A formulation as Mixed Integer Programming problem is presented and proved effective for obtaining bounds. Preliminary computational results are reported. The final part of this work outlines some future research directions.

Cited by 0

Using subgoal chaining to address the local minimum problem

Jonathan P. Lewis · St Andrews Research Repository (St Andrews Research Repository) · 2002

A common problem in the area of non-linear function optimisation is that of not being able to guarantee finding the global optimum of the function in a feasible time especially when local optima exist. This problem applies to various areas of heuristic search. One of these areas concerns standard training techniques for feedforward neural networks. The element of heuristic search consists of attempting to find a neural weight state corresponding to the lowest training error. This problem may be termed the local minimum problem. The local minimum problem is addressed for feedforward neural networks. This is done by first establishing the conditions under which local minimum interference for the training process is to be expected. A target based approach to subgoal chaining in supervised learning is then investigated. This is a method to improve travel for neural networks by directing it more precisely through local subgoals than may be achieved through a more distant goal. It is shown however that linear subgoal chains are not sufficient to overcome the local minimum problem. Two novel training techniques are presented which use non-linear subgoal chains and are examined for their capability to address the local minimum problem. It is found that attempting to target a neural network to do something it cannot may lead to suboptimal training. It is also found that targeting a network to do something it is capable of generally leads to successful training. A novel system is presented which is designed to create optimal realisable targets for unrealisable goals. This allows neural networks to subsequently achieve the optimal weight state through a sufficiently powerful training method such as subgoal chaining. The results are shown to be consistent with the theoretical expectations.

Cited by 5

Local porosity of the free boundary in a minimum problem

Yuwei Hu, Jun Zheng · Electronic Research Archive · 2023

<abstract><p>Given certain set $ \mathcal{K} $ and functions $ q $ and $ h $, we study geometric properties of the set $ \partial\{x\in\Omega:u(x) > 0\} $ for non-negative minimizers of the functional $ \mathcal{J} (u) = \int_{\Omega }^{} \, \left(\frac{1}{p}| \nabla u| ^p+q(u^+)^\gamma +hu\right)\text{d}x $ over $ \mathcal{K} $, where $ {\Omega \subset} \mathbb{R} ^n(n\geq 2) $ is an open bounded domain, $ p\in(1, +\infty) $ and $ \gamma \in (0, 1] $ are constants, $ u^+ $ is the positive part of $ u $ and $ \partial\{x\in\Omega :u(x) > 0\} $ is the so-called free boundary. Such a minimum problem arises in physics and chemistry for $ \gamma = 1 $ and $ \gamma \in(0, 1) $, respectively. Using the comparison principle of $ p $-Laplacian equations, we establish first the non-degeneracy of non-negative minimizers near the free boundary, then prove the local porosity of the free boundary.</p></abstract>

Cited by 0Open access

Adaptively Solving the Local-Minimum Problem for Deep Neural Networks

Huachuan Wang, James Lo · arXiv (Cornell University) · 2021

This paper aims to overcome a fundamental problem in the theory and application of deep neural networks (DNNs). We propose a method to solve the local minimum problem in training DNNs directly. Our method is based on the cross-entropy loss criterion's convexification by transforming the cross-entropy loss into a risk averting error (RAE) criterion. To alleviate numerical difficulties, a normalized RAE (NRAE) is employed. The convexity region of the cross-entropy loss expands as its risk sensitivity index (RSI) increases. Making the best use of the convexity region, our method starts training with an extensive RSI, gradually reduces it, and switches to the RAE as soon as the RAE is numerically feasible. After training converges, the resultant deep learning machine is expected to be inside the attraction basin of a global minimum of the cross-entropy loss. Numerical results are provided to show the effectiveness of the proposed method.

Cited by 1Open access

An iterated local search algorithm for the minimum differential dispersion problem

Yangming Zhou, Jin‐Kao Hao · arXiv (Cornell University) · 2016

Given a set of $n$ elements separated by a pairwise distance matrix, the minimum differential dispersion problem (Min-Diff DP) aims to identify a subset of m elements (m < n) such that the difference between the maximum sum and the minimum sum of the inter-element distances between any two chosen elements is minimized. We propose an effective iterated local search (denoted by ILS_MinDiff) for Min-Diff DP. To ensure an effective exploration and exploitation of the search space, the proposed ILS_MinDiff algorithm iterates through three sequential search phases: a fast descent-based neighborhood search phase to find a local optimum from a given starting solution, a local optima exploring phase to visit nearby high-quality solutions around a given local optimum, and a local optima escaping phase to move away from the current search region. Experimental results on six data sets of 190 benchmark instances demonstrate that ILS_MinDiff competes favorably with the state-of-the-art algorithms by finding 130 improved best results (new upper bounds).

Cited by 0Open access

Improved artificial potential field to solve the problem of local minimum

Hao Jiang · Applied and Computational Engineering · 2023

The artificial potential field approach is a traditional method that is frequently used for local path planning, but in order for it to be successful, it requires more than unreachable target sites, local minimums, and oscillation difficulties. An enhanced method that is based on discretized sampling was presented as a potential solution to these issues as a result of this information. In the first place, a decision must be made regarding whether or not to enter the local minimum mark. In order to free ourselves from the confines of the local minimum, we subsequently devised a motion that was perpendicular to the direction in which the object was being moved. In the end, a regression search is carried out in order to further hone in on the correct path. The results of the simulation show that this method is capable of efficiently handling the local minimum problem and providing appropriate paths, as demonstrated by a comparison between the conventional method and the enhanced method.

Cited by 1Open access

NuMWVC: A Novel Local Search for Minimum Weighted Vertex Cover Problem

Ruizhi Li, Shaowei Cai, Shuli Hu et al. · Proceedings of the AAAI Conference on Artificial Intelligence · 2018

The minimum weighted vertex cover (MWVC) problem is a well known combinatorial optimization problem with important applications. This paper introduces a novel local search algorithm called NuMWVC for MWVC based on three ideas. First, four reduction rules are introduced during the initial construction phase. Second, the configuration checking with aspiration is proposed to reduce cycling problem. Moreover, a self-adaptive vertex removing strategy is proposed to save time.

Cited by 8Open access

An improved artificial potential field method for solving local minimum problem

Qing Li, Lijun Wang, Bo Chen et al. · 2011

An improved artificial potential field (APF) method is proposed for solving local minimum problems. An additional force is provided and the virtual local minimum area is developed. The influences of parameters in improved APF are also discussed and the genetic algorithm (GA) is introduced for parameters optimization. Some simulation studies with single obstacle and multiple obstacles have carried out to demonstrate the effectiveness of the proposed algorithms.

Cited by 33

On power sums of convex functions in local minimum energy problem

Kanya Ishizaka · Journal of Mathematical Inequalities · 2012

In this paper, new inequalities on the power sums of a convex function are derived and the monotonically decreasing nature of the Riemann sum of a function including a certain strong convexity is shown. It is also shown that a derived inequality has a direct implication in a local minimum energy problem in two-dimensional hexagonal packing.

Cited by 2Open access

Minimum Cost Source Location Problem with Local 3-Vertex-Connectivity Requirements ⋆

Toshimasa Ishii A · 2008

Let G = (V, E) be a simple undirected graph with a set V of vertices and a set E of edges. Each vertex v ∈ V has a demand d(v) ∈ Z+ and a cost c(v) ∈ R+, where Z+ and R+ denote the set of nonnegative integers and the set of nonnegative reals, respectively. The source location problem with vertex-connectivity requirements in a given graph G asks to find a set S of vertices minimizing � v∈S c(v) such that there are at least d(v) pairwise vertex-disjoint paths from S to v for each vertex v ∈ V − S. It is known that if there exists a vertex v ∈ V with d(v) ≥ 4, then the problem is NP-hard even in the case where every vertex has a uniform cost. In this paper, we show that the problem can be solved in O(|V | 4 log 2 |V |) time if d(v) ≤ 3 holds for each vertex v ∈ V. Key words: graph algorithm, undirected graph, location problem, local vertex-connectivity, polynomial time algorithm

Cited by 0

NuMDS: An Efficient Local Search Algorithm for Minimum Dominating Set Problem

Rui Sun, Zhaohui Liu, Yiyuan Wang et al. · 2024

The minimum dominating set (MDS) problem is a crucial NP-hard combinatorial optimization problem with wide applications in real-world scenarios. In this paper, we propose an efficient local search algorithm namely NuMDS to solve the MDS, which comprises three key ideas. First, we introduce a dominate propagation-based reduction method that fixes a portion of vertices in a given graph. Second, we develop a novel two-phase initialization method based on the decomposition method. Third, we propose a multi-stage local search procedure, which adopts three different search manners according to the current stage of the search. We conduct extensive experiments to demonstrate the outstanding effectiveness of NuMDS, and the results clearly indicate that NuMDS outperforms previous state-of-the-art algorithms on almost all instances.

Cited by 2Open access