Finding sparse solutions to a system of equations and/or inequalities is an important topic in many application areas such as signal processing, statistical regression and nonparametric modeling. Various continuous relaxation models have been proposed and widely studied to deal with the discrete nature of the underlying problem. In this paper, we propose a quadratically constrained [Formula: see text] (0 < q < 1) minimization model for finding sparse solutions to a quadratic system. We prove that solving the proposed model is strongly NP-hard. To tackle the computation difficulty, a first order necessary condition for local minimizers is derived. Various properties of the proposed model are studied for designing an active-set-based descent algorithm to find candidate solutions satisfying the proposed condition. In addition to providing a theoretical convergence proof, we conduct extensive computational experiments using synthetic and real-life data to validate the effectiveness of the proposed algorithm and to show the superior capability in finding sparse solutions of the proposed model compared with other known models in the literature. We also extend our results to a quadratically constrained [Formula: see text] (0 < q < 1) minimization model with multiple convex quadratic constraints for further potential applications. Summary of Contribution: In this paper, we propose and study a quadratically constrained [Formula: see text] minimization (0 < q < 1) model for finding sparse solutions to a quadratic system which has wide applications in sparse signal recovery, image processing and machine learning. The proposed quadratically constrained [Formula: see text] minimization model extends the linearly constrained [Formula: see text] and unconstrained [Formula: see text]-[Formula: see text] models. We study various properties of the proposed model in aim of designing an efficient algorithm. Especially, we propose an unrelaxed KKT condition for local/global minimizers. Followed by the properties studied, an active-set based descent algorithm is then proposed with its convergence proof being given. Extensive numerical experiments with synthetic and real-life Sparco datasets are conducted to show that the proposed algorithm works very effectively and efficiently. Its sparse recovery capability is superior to that of other known models in the literature.

Optimal reliability design (ORD) problem is challenging and fundamental to the study of system reliability. For a system with u components/stages where each of them can be set in m possible reliability levels, state-of-the-art linear reformulation models of ORD problem require O(um) binary variables, O(mn) continuous variables together with either O(mn) inequality constraints or O(um) equality constraints. Using the special property of prime factorization and adopting the logarithmic expression technique, in this article, we propose a novel linear reformulation model of the ORD problem requiring O(um) binary variables, O(mn n! ) continuous variables, and very few linear constraints. This theoretic reduction in variables and constraints can lead to significant savings in computational efforts. Our numerical experiments further confirm the drastic reduction in computational time for solving ORD problems in large size.

Polynomial discrete programming problems are commonly faced but hard to solve. Treating the nonconvex cross-product terms is the key. State-of-the-art methods usually convert such a problem into a 0-1 mixed-integer linear programming problem and then adopt the branch-and-bound scheme to find an optimal solution. Much effort has been spent on reducing the required numbers of variables and linear constraints as well as on avoiding unbalanced branch-and-bound trees. This study presents a set of equations that linearize the discrete cross-product terms in an extremely effective manner. It is shown that embedding the proposed “equations for linearizing discrete products” into those state-of-the-art methods in the literature not only significantly reduces the required number of linear constraints from O(h3n3) to O(hn) for a cubic polynomial discrete program with n variables in h possible values but also tighten these methods with much more balanced branch-and-bound trees. Numerical experiments confirm a two-order (102-times) reduction in computational time for some randomly generated cubic polynomial discrete programming problems. There is a Video Overview associated with this article, available as supplemental material.

The triangular norm-based operations in fuzzy logic usually lead to non-L-R type fuzzy sets. This study considers mathematical programming problems with non-L-R type fuzzy parameters. It shows that the fuzzy solutions to such problems can be obtained by solving an optimization problem on a mixed domain. The necessary and sufficient conditions for solving the resulting optimization problems are investigated by employing the theory of convex optimization on mixed domains. This is the first attempt to solve the fuzzy optimization problem with non-L-R type membership functions in view of optimization problems on a mixed domain.

This paper studies the classical task assignment problem (TAP) in which M unbreakable tasks are assigned to N agents with the objective to minimize the communication and process costs subject to each agent's capacity constraint. Because a large-size TAP involves many binary variables, most, if not all, traditional methods experience the difficulty in solving the problem within a reasonable time period. Recent works present a logarithmic approach to reduce the number of binary variables in problems with mixed-integer variables. This study proposes a new logarithmic method that significantly reduces the numbers of binary variables and inequality constraints in solving task assignment problems. Our numerical experiments demonstrate that the proposed method is superior to other known methods of this kind for solving large-size TAPs.

This paper considers the latticized linear optimization (LLO) problem and its variants, which are a special class of optimization problems constrained by fuzzy relational equations or inequalities. We show that an optimal solution to such a problem can be obtained in polynomial time as long as the objective function is a max-separable function with continuous monotone components. We further show that the set of all optimal solutions is fully determined by one maximum optimal solution and a finite number of minimal optimal solutions. The maximum optimal solution can be constructed in polynomial time once the optimal objective value is known, while the detection of all minimal optimal solutions in an efficient manner remains as a challenging problem. The relation between LLO and max-separable optimization and related issues are also investigated.

The performance of the fuzzy-controlled curved search method depends heavily on the membership functions of the categories of output error and step length used in the controller. We present a fuzzy neuron controller of backpropagation training algorithms for neural networks. With the objective of reducing tuning time, we introduce a self-adaptive fuzzy neuron controller for determining the size of step length. Computational results indicate a strong potential for reduction in tuning time for the fuzzy neuron controller in the neural network learning.

This paper presents a neural-network computational scheme with time-delay consideration for solving convex quadratic programming problems. Based on some known results, a delay margin is explicitly determined for the stability of the neural dynamics, under which the states of the neural network does not oscillate. The configuration of the proposed neural network is provided. Operational characteristics of the neural network are demonstrated via numerical examples.

In this paper we present a new learning method of back propagation training algorithms for neural networks. With the objective of reducing training time, we introduce a curved search method with a fixed step size for determining weight adjustments. Computational results indicate a strong potential for significant reduction in training time.

This paper deals with systems of fuzzy inequalities. It shows that a system of fuzzy inequalities with piecewise linear membership functions can be converted to a one-constraint nonlinear programming problem by employing the concepts of surrogate constraints and maximum entropy. An augmented Lagrangean algorithm is then applied to solve the resulting problem. Some computational results are included.

We study a system of fuzzy relation equations with max-product composition and present an efficient solution procedure to characterize the whole solution set by finding the maximum solution as well as the complete set of minimal solutions. Instead of solving the problem combinatorially, the procedure identifies the "nonminimal" solutions and eliminates them from the set of minimal solutions.

The due-date bargainer is a useful tool to support negotiation on due dates between a manufacturer and its customers. To improve the computational performance of an earlier version of the due-date bargainer, we present a new soft computing approach. It uses a genetic algorithm to find the best priority sequence of customer orders for resource allocation, and fuzzy logic operations to allocate the resources and determine the order completion times, following the priority sequence of orders. To extend the due-date bargainer to accommodate bargaining with several customers at the same time, we propose a method to distribute the total penalty using marginal penalties for the individual bargainers. A demonstration software package implementing the improved due-date bargainer has been developed. It is targeted at apparel manufacturing enterprises. Experiments using realistic resource data and randomly generated orders have achieved satisfactory results.

Solving systems of fuzzy relation equations is an important topic in fuzzy set theory. This paper studies the composite interval-valued fuzzy relation equations. After analyzing the properties of its solution set, we convert the fuzzy relation equations into a fuzzy relation inequality system and propose an efficient computational procedure to generate the whole solution set. Examples are included to illustrate the idea and algorithm.

For a make-to-order manufacturing system, the uncertainty and flexibility of the due dates required by customers and the production capacity owned by the manufacturer can be modeled in a fuzzy environment. A combined due-date assignment and production planning methodology for the make-to-order manufacturing systems is developed. The fuzzy approach determines the optimal due dates for the manufacturer based upon a "rough-cut" resource balance, while a customer can request earlier due dates by paying a higher price to cover the extra manufacturing cost incurred. The resulting fuzzy due-date bargainer is exercised using manufacturing resource planning (MRP-II) data from a furniture manufacturing company. Experimental results indicate its potential as a useful tool for real applications.

Due to the nondeterministic nature of the business environment of a manufacturing enterprise, it is more appropriate to describe the aggregated production planning by using a fuzzy mathematical programming model. In this paper, a genetics-based inexact approach is proposed to imitate the human decision procedure for production planning. Instead of locating one exact optimal solution, the proposed approach finds a family of inexact solutions within an acceptable level by adopting a mutation operator to move along a weighted gradient direction. Then, a decision maker can select a preferred solution by examining a convex combination of the solutions in the family via the human-computer interaction. Our computational experiments illustrate how the enterprise managers can be more satisfied by this new approach than others.