Basic Optimization Problem Research Articles

A Tight Linear Time (1/2)-Approximation for Unconstrained Submodular Maximization

We consider the \sf Unconstrained Submodular Maximization problem in which we are given a nonnegative submodular function $f:2^{\mathcal{N}}\rightarrow \mathbb{R}^+$, and the objective is to find a subset $S\subseteq \mathcal{N}$ maximizing $f(S)$. This is one of the most basic submodular optimization problems, having a wide range of applications. Some well-known problems captured by \sf Unconstrained Submodular Maximization include \sf Max-Cut, \sf Max-DiCut, and variants of \sf Max-SAT and maximum facility location. We present a simple randomized linear time algorithm achieving a tight approximation guarantee of 1/2, thus matching the known hardness result of Feige, Mirrokni, and Vondrák [SIAM J. Comput., 40 (2011), pp. 1133--1153]. Our algorithm is based on an adaptation of the greedy approach which exploits certain symmetry properties of the problem.

Hierarchical Control Strategy based on Robust MPC and Integral Sliding mode - Application to a Continuous Photobioreactor

This paper proposes the design of a hierarchical control strategy formed by a two-level controller: a Linearized Robust MPC (LRMPC) and an Integral Sliding Mode (ISM) control laws. The proposed strategy guarantees robustness towards parameters mismatch for a macroscopic continuous photobioreactor model, obtained from mass balance based modelling. Firstly, as a starting point, this work focuses on classical robust nonlinear model predictive control law under model parameters uncertainties implying solving a basic min-max optimization problem for setpoint trajectory tracking. We reduce this problem into a regularized optimization problem based on the use of linearization techniques, to ensure a good trade-off between tracking accuracy and computation time. Secondly, in order to eliminate the static error due to the fact that the nonlinear model is approximated through linearization in the LRMPC law, an ISM controller is synthesized relying on the knowledge of the nonlinear model of the system. Finally, the efficiency of the developed hierarchical approach is illustrated through numerical results and robustness against parameter uncertainties is discussed for the worst case model mismatch.

Discrete convexity and polynomial solvability in minimum 0-extension problems

A $$0$$ -extension of graph $$\varGamma $$ is a metric $$d$$ on a set $$V$$ containing the vertex set $$V_{\varGamma }$$ of $$\varGamma $$ such that $$d$$ extends the shortest path metric of $$\varGamma $$ and for all $$x \in V$$ there exists a vertex $$s$$ in $$\varGamma $$ with $$d(x,s) = 0$$ . The minimum $$0$$ -extension problem 0-Ext $$[\varGamma ]$$ on $$\varGamma $$ is: given a set $$V \supseteq V_{\varGamma }$$ and a nonnegative cost function $$c$$ defined on the set of all pairs of $$V$$ , find a $$0$$ -extension $$d$$ of $$\varGamma $$ with $$\sum _{xy}c(xy) d(x,y)$$ minimum. The $$0$$ -extension problem generalizes a number of basic combinatorial optimization problems, such as minimum $$(s,t)$$ -cut problem and multiway cut problem. Karzanov proved the polynomial solvability of 0-Ext $$[\varGamma ]$$ for a certain large class of modular graphs $$\varGamma $$ , and raised the question: What are the graphs $$\varGamma $$ for which 0-Ext $$[\varGamma ]$$ can be solved in polynomial time? He also proved that 0-Ext $$[\varGamma ]$$ is NP-hard if $$\varGamma $$ is not modular or not orientable (in a certain sense). In this paper, we prove the converse: if $$\varGamma $$ is orientable and modular, then 0-Ext $$[\varGamma ]$$ can be solved in polynomial time. This completes the classification of graphs $$\varGamma $$ for which 0-Ext $$[\varGamma ]$$ is tractable. To prove our main result, we develop a theory of discrete convex functions on orientable modular graphs, analogous to discrete convex analysis by Murota, and utilize a recent result of Thapper and Živný on valued CSP.

An alternative proof of a PTAS for fixed-degree polynomial optimization over the simplex

The problem of minimizing a polynomial over the standard simplex is one of the basic NP-hard nonlinear optimization problems—it contains the maximum clique problem in graphs as a special case. It is known that the problem allows a polynomial-time approximation scheme (PTAS) for polynomials of fixed degree, which is based on polynomial evaluations at the points of a sequence of regular grids. In this paper, we provide an alternative proof of the PTAS property. The proof relies on the properties of Bernstein approximation on the simplex. We also refine a known error bound for the scheme for polynomials of degree three. The main contribution of the paper is to provide new insight into the PTAS by establishing precise links with Bernstein approximation and the multinomial distribution.

Concurrent conditional clustering of multiple networks: COCONETS.

The accumulation of high-throughput data from different experiments has facilitated the extraction of condition-specific networks over the same set of biological entities. Comparing and contrasting of such multiple biological networks is in the center of differential network biology, aiming at determining general and condition-specific responses captured in the network structure (i.e., included associations between the network components). We provide a novel way for comparison of multiple networks based on determining network clustering (i.e., partition into communities) which is optimal across the set of networks with respect to a given cluster quality measure. To this end, we formulate the optimization-based problem of concurrent conditional clustering of multiple networks, termed COCONETS, based on the modularity. The solution to this problem is a clustering which depends on all considered networks and pinpoints their preserved substructures. We present theoretical results for special classes of networks to demonstrate the implications of conditionality captured by the COCONETS formulation. As the problem can be shown to be intractable, we extend an existing efficient greedy heuristic and applied it to determine concurrent conditional clusters on coexpression networks extracted from publically available time-resolved transcriptomics data of Escherichia coli under five stresses as well as on metabolite correlation networks from metabolomics data set from Arabidopsis thaliana exposed to eight environmental conditions. We demonstrate that the investigation of the differences between the clustering based on all networks with that obtained from a subset of networks can be used to quantify the specificity of biological responses. While a comparison of the Escherichia coli coexpression networks based on seminal properties does not pinpoint biologically relevant differences, the common network substructures extracted by COCONETS are supported by existing experimental evidence. Therefore, the comparison of multiple networks based on concurrent conditional clustering offers a novel venue for detection and investigation of preserved network substructures.

A modified harmony search method for environmental/economic load dispatch of real-world power systems

Economic load dispatch (ELD) problem is one of the basic and important optimization problems in a power system. However, considering practical constraints of real-world power systems such as ramp rate limits, prohibited operating zones, valve loading effects, multi-fuel options, spinning reserve and transmission system losses in ELD problem makes it a non-convex optimization problem, which is a challenging one and cannot be solved by traditional methods. Moreover, considering environmental issues, results in combined economic and emission load dispatch (CEELD) problem that is a multiobjective optimization model with two non-commensurable and contradictory objectives. In this paper, a modified harmony search algorithm (MHSA) proposed and applied to solve ELD and CEELD problem considering the abovementioned constraints. In the proposed MHSA, a new improvising method based on wavelet mutation together with a new memory consideration scheme based on the roulette wheel mechanism are proposed which improves the accuracy, convergence speed, and robustness of the classical HSA. Performance of the proposed algorithm is investigated by applying it to solve various test systems having non-convex solution spaces. To Show the effectiveness of the proposed method, obtained results compared with classical harmony search algorithm (HSA) and some of the most recently published papers in the area.

Structural Optimization Design Considering Reliability Constraints

In engineering applications the uncertainties of the structural parameters are inherent and the scatter from their nominal ideal values is in most cases unavoidable. These uncertainties play a dominant role in structural performance and the reliability-based design optimization is a useful method to assess the uncertainty influence. Compared to the basic deterministic-based optimization problem, the latter considers additional non-deterministic constraint functions and will provide the structure more safety. This paper proposed a Fourier orthogonal neural network method to the structural reliability analysis and reliability-based optimization considering uncertainties, the main aim is to minimize the weight of the structure under certain reliability constraints, and to obtain economic benefit meanwhile ensure the safety of the structure.

An adaptive amoeba algorithm for constrained shortest paths

The constrained shortest path problem (CSP) is one of the basic network optimization problems, which plays an important part in real applications. In this paper, an adaptive amoeba algorithm is combined with the Lagrangian relaxation algorithm to solve the CSP problem. The proposed method is divided into two steps: (1) the adaptive amoeba algorithm is modified to solve the shortest path problem (SPP) in a directed network; (2) the modified adaptive amoeba algorithm is combined with the Lagrangian relaxation method to solve the CSP problem. In addition, the evolving processes of the adaptive amoeba model have been detailed in the paper. Two examples are used to illustrate the efficiency of the proposed method. The results show that the proposed method can deal with the CSP problem effectively.

A hybridized tabu search approach for the minimum weight vertex cover problem

The minimum weight vertex cover problem is a basic combinatorial optimization problem defined as follows. Given an undirected graph and positive weights for all vertices the objective is to determine a subset of the vertices which covers all edges such that the sum of the related cost values is minimized. In this paper we apply a modified reactive tabu search approach for solving the problem. While the initial concept of reactive tabu search involves a random walk we propose to replace this random walk by a controlled simulated annealing. Numerical results are presented outperforming previous metaheuristic approaches in most cases.

Minimize Traffic Congestion: An Application of Maximum Flow in Dynamic Networks

Minimize Traffic Congestion: An Application of Maximum Flow in Dynamic NetworksAn important characteristic of a network is its capacity to carry flow. What, given capacities on the arcs, is the maximum flow that can be sent between any two nodes? The dynamic version of the maximum flow problem on networks that generalizes the well-known static one. This basic combinatorial optimization problem has a large implementation for many practical problems. Traffic congestion is a consequence of the nature of supply and demand: capacity is time consuming and costly to build and is fixed for long time periods, demand fluctuates over time, and transport services cannot be stored to smooth imbalances between capacity and demand. In this paper, I tried to solve the traffic congestion problem i.e. Maximum flow of goods in a dynamic network with the help of a Lingo Model. The same can be generalized for the large product if the software supports the systems.

Evolutionary kernel machines

This special issue on evolutionary kernel machines is dedicated to the latest developments in combining kernelbased algorithms and evolutionary computation. Kernelbased methods are among the state-of-the-art approaches in machine learning and data mining (e.g., [3, 4]). The idea of kernel machines is to change the representation of the input data by mapping the input space to some metric feature space. The kernel is a real-valued function of two input space elements that corresponds to a scalar product of its arguments mapped to the feature space [1]. In algorithms using the ‘kernel trick’, all operations in the feature space can be expressed by scalar products—and the kernel replaces all these operations. What are the advantages of using a kernel? The kernel renders an explicit mapping of input data to the feature space unnecessary, and a kernel function can usually be computed much more efficiently than mapping two data points to the feature space and computing their scalar product in the feature space representation. Thus a kernel allows to work in very highdimensional feature spaces. Further, kernel-based algorithms are very modular and flexible. Apart from hyperparameters (such as learning rates and regularization parameters) only the kernel function has to be changed when the problem domain changes. The kernel defines the feature space and the algorithm’s notion of similarity between input objects. The clear interface makes handling of non-standard data (e.g., biological sequences) comparatively easy. Finally, the convenient mathematical structure of the kernelinduced feature and hypothesis spaces supports design and analysis of the adaptive system. The basic optimization problems arising when applying kernel methods—for example, training a support vector machine—are usually solved using mathematical programming. However, in some scenarios, there are good reasons to employ stochastic optimization techniques due to the need to solve problems with local optima, high dimensionality, discontinuities, multiple conflicting objectives, and noise. Examples are the fundamental task of finding a proper kernel function for an application (e.g., [2]), feature selection, or label assignment in semi-supervised learning. Here, evolutionary algorithms come into play, which are applicable to objective functions suffering from the aforementioned characteristics. In addition, they are embarrassingly parallelizable. This special issue comprises three original research papers. The article by Koch, Bischl, Flasch, Bartz-Beielstein and Konen studies the problem of automated tuning of support vector machines. It shows how kernels can be evolved by genetic programming, and how sophisticated parameter tuning methods can help to find appropriate hyperparameters. Glasmachers, Koutnik and Schmidhuber introduce a kernel-based evolutionary framework for learning continuous functions. Within this framework, nested function spaces of increasing complexity are used to incrementally determine an appropriate level of complexity. Finally, Gieseke, Kramer, Airola and Pahikkala employ stochastic local search schemes to address the combinatorial problem of assigning labels to unlabeled data in semiand unsupervised kernel-based classification. O. Kramer (&) Oldenburg, Germany e-mail: oliver.kramer@uni-oldenburg.de

Hybrid genetic algorithm and augmented neural network application for solving the online advertisement scheduling problem with contextual targeting

Worldwide growth of the online community continues to push the popularity of internet marketing. Fueled by this trend, the online advertising industry is experiencing unprecedented revenue growth. One of the most important drivers of this revenue is banner advertising, which has long been a staple of the online advertising industry. Previous research has introduced quantitative models and solution approaches for the challenging basic scheduling optimization problem. We extend this work by incorporating the most common and popular trend in the in the industry, online advertisement targeting. In addition, motivated by the NP-hard nature of the resulting problem, we propose and test several heuristic and metaheuristic based solution techniques for the proposed problem.

Decorous Lower Bounds for Minimum Linear Arrangement

Minimum linear arrangement is a classical basic combinatorial optimization problem from the 1960s that turns out to be extremely challenging in practice. In particular, for most of its benchmark instances, even the order of magnitude of the optimal solution value is unknown, as testified by the surveys on the problem that contain tables in which the best-known solution value often has one more digit than the best-known lower bound value. In this paper, we propose a linear programming-based approach to compute lower bounds on the optimum. This allows us, for the first time, to show that the best-known solutions are indeed not far from optimal for most of the benchmark instances.

Imprecision and Flexible Constraints in Fuzzy Vehicle Routing Problem

The Vehicle Routing Problem (VRP) is a well-known combinatorial optimization problem. The classical definitions of vehicle routing problems often lack handling of uncertain parameters and flexibility of constraints. The most popular approaches to these aspects involve probability distributions for uncertainty and penalty-based goal function for flexibility. In this paper a different approach is proposed which involves fuzzy sets framework, modeling both imprecision and flexibility, and analyzing practical solution approaches for VRP with uncertain information. Several variants of the basic problem and fuzzy optimization problem formulations are described. The fuzzy VRP is obtained when some of the elements in the formulation are fuzzy. The main versions of VRP that have appeared in the literature are reviewed and the standard approaches for solving the corresponding models are described.

Modeling agents based on aspiration adaptation theory

Creating agents that realistically simulate and interact with people is an important problem. In this paper we present strong empirical evidence that such agents should be based on bounded rationality, and specifically on key elements from Aspiration Adaptation Theory (AAT). First, we analyzed the strategies people described they would use to solve two relatively basic optimization problems involving one and two parameters. Second, we studied the agents a different group of people wrote to solve these same problems. We then studied two realistic negotiation problems involving five and six parameters. Again, first we studied the negotiation strategies people used when interacting with other people. Then we studied two state of the art automated negotiation agents and negotiation sessions between these agents and people. We found that in both the optimizing and negotiation problems the overwhelming majority of automated agents and people used key elements from AAT, even when optimal solutions, machine learning techniques for solving multiple parameters, or bounded techniques other than AAT could have been implemented. We discuss the implications of our findings including suggestions for designing more effective agents for game and simulation environments.

Two complexity results on c-optimality in experimental design

Finding a c-optimal design of a regression model is a basic optimization problem in statistics. We study the computational complexity of the problem in the case of a finite experimental domain. We formulate a decision version of the problem and prove its $\boldsymbol{\mathit{NP}}$ -completeness. We provide examples of computationally complex instances of the design problem, motivated by cryptography. The problem, being $\boldsymbol{\mathit{NP}}$ -complete, is then relaxed; we prove that a decision version of the relaxation, called approximate c-optimality, is P-complete. We derive an equivalence theorem for linear programming: we show that the relaxed c-optimality is equivalent (in the sense of many-one LOGSPACE-reducibility) to general linear programming.

Lower Bounds for the Minimum Linear Arrangement of a Graph

Abstract Minimum Linear Arrangement is a classical basic combinatorial optimization problem from the 1960s, which turns out to be extremely challenging in practice. In particular, for most of its benchmark instances, even the order of magnitude of the optimal solution value is unknown, as testified by the surveys on the problem that contain tables in which the best known solution value often has one more digit than the best known lower bound value. In this paper, we propose a linear-programming based approach to compute lower bounds on the optimum. This allows us, for the first time, to show that the best known solutions are indeed not far from optimal for most of the benchmark instances.

Towards an integrated framework for supply chain management in the batch chemical process industry

There is a large body of work on supply chain (SC) optimization in the chemical process industry (CPI). However, some of the basic aspects of the optimization problem are not adequately handled by the models and solution strategies developed so far in the literature. This paper focuses on the underlying philosophy of our approach to supply chain management (SCM) in the CPI, which aims to overcome the challenges posed by this problem. Two main topics that offer great opportunities for improvement in SCM are discussed. These are the development of modeling approaches and solution strategies that reflect SC dynamics, the inclusion of environmental considerations, and the incorporation of novel business aspects and key performance indicators (KPI) into the existing formulations to enlarge the scope of SC analysis, which is currently rather limited. Our integrated solution strategy for SCM, which covers the aforementioned aspects and implements the ideas and concepts developed in our research, is also presented and its advantages are highlighted in a case study.

Computing Minimum Cuts by Randomized Search Heuristics

We study the minimum s-t-cut problem in graphs with costs on the edges in the context of evolutionary algorithms. Minimum cut problems belong to the class of basic network optimization problems that occur as crucial subproblems in many real-world optimization problems and have a variety of applications in several different areas. We prove that there exist instances of the minimum s-t-cut problem that cannot be solved by standard single-objective evolutionary algorithms in reasonable time. On the other hand, we develop a bi-criteria approach based on the famous maximum-flow minimum-cut theorem that enables evolutionary algorithms to find an optimal solution in expected polynomial time.

Visualization approaches for the prototype improvement problem

AbstractOne of the basic engineering optimization problems is improvement of the prototype. This problem is often encountered by industrial and academic organizations that produce and design various objects (e.g. motor vehicles, machine tools, ships, and aircrafts). This paper presents an approach for improving the prototype by constructing the feasible and Pareto sets while performing multicriteria analysis. We introduce visualization methods that facilitate construction of the feasible and Pareto sets. Using these techniques developed on the basis of Parameter Space Investigation method, an expert can correctly state and solve the problem under consideration in a series of dialogues with the computer. Finally, we present a case study of improving the ship prototype. Copyright © 2008 John Wiley & Sons, Ltd.