Solution Of Subproblems Research Articles

Primal convergence from dual subgradient methods for convex optimization

When solving a convex optimization problem through a Lagrangian dual reformulation subgradient optimization methods are favorably utilized, since they often find near-optimal dual solutions quickly. However, an optimal primal solution is generally not obtained directly through such a subgradient approach unless the Lagrangian dual function is differentiable at an optimal solution. We construct a sequence of convex combinations of primal subproblem solutions, a so called ergodic sequence, which is shown to converge to an optimal primal solution when the convexity weights are appropriately chosen. We generalize previous convergence results from linear to convex optimization and present a new set of rules for constructing the convexity weights that define the ergodic sequence of primal solutions. In contrast to previously proposed rules, they exploit more information from later subproblem solutions than from earlier ones. We evaluate the proposed rules on a set of nonlinear multicommodity flow problems and demonstrate that they clearly outperform the ones previously proposed.

Scalable methods to integrate task knowledge with the Three-Weight Algorithm for hybrid cognitive processing via optimization

Abstract In this paper we consider optimization as an approach for quickly and flexibly developing hybrid cognitive capabilities that are efficient, scalable, and can exploit task knowledge to improve solution speed and quality. Given this context, we focus on the Three-Weight Algorithm, which is interruptible, scalable, and aims to solve general optimization problems. We propose novel methods by which to integrate diverse forms of task knowledge with this algorithm in order to improve expressiveness, efficiency, and scaling across a variety of problems. To demonstrate these techniques, we focus on two large-scale constraint-satisfaction domains, Sudoku and circle packing. In Sudoku, we efficiently and dynamically integrate knowledge of logically deduced sub-problem solutions; this integration leads to improved system reactivity and greatly reduced solution time for large problem instances. In circle packing, we efficiently integrate knowledge of task dynamics, as well as real-time human guidance via mouse gestures; these integrations lead to greatly improved system reactivity, as well as world-record-breaking solutions on very large packing problems. These results exemplify how cognitive architecture can integrate high-level knowledge with powerful optimization techniques in order to effectively and efficiently contend with a variety of cognitive tasks.

Parallel-machine scheduling to minimize overtime and waste costs

This article considers scheduling products in a parallel, non-identical machine environment subject to sequence-dependent setup costs and sequence-dependent setup times, where production waste and processing time of a product depend on feasible machine assignments. A Mixed-Integer Programming (MIP) formulation is developed that captures trade-offs between overtime labor costs and waste costs. Two solution procedures are developed to address large problem instances. One procedure is an algorithm that determines a vector of product-to-machine assignments that assists an MIP solver to find an initial feasible solution. The second procedure is a decomposition heuristic that iteratively solves a relaxed subproblem and uses the subproblem solution to fix assignment variables in the main MIP formulation. In addition, bounds on the quality of solutions found using the decomposition heuristic are presented. Experiments are conducted that show the developed formulation outperforms more traditional scheduling objectives with respect to the waste and overtime labor costs. Additional experimentation investigates the effects that problem parameter values have on total waste and labor costs, performance of the approaches, and the use of overtime labor.

MOEA/D with uniform decomposition measurement for many-objective problems

Many-objective problems (MAPs) have put forward a number of challenges to classical Pareto-dominance based multi-objective evolutionary algorithms (MOEAs) for the past few years. Recently, researchers have suggested that MOEA/D (multi-objective evolutionary algorithm based on decomposition) can work for MAPs. However, there exist two difficulties in applying MOEA/D to solve MAPs directly. One is that the number of constructed weight vectors is not arbitrary and the weight vectors are mainly distributed on the boundary of weight space for MAPs. The other is that the relationship between the optimal solution of subproblem and its weight vector is nonlinear for the Tchebycheff decomposition approach used by MOEA/D. To deal with these two difficulties, we propose an improved MOEA/D with uniform decomposition measurement and the modified Tchebycheff decomposition approach (MOEA/D-UDM) in this paper. Firstly, a novel weight vectors initialization method based on the uniform decomposition measurement is introduced to obtain uniform weight vectors in any amount, which is one of great merits to use our proposed algorithm. The modified Tchebycheff decomposition approach, instead of the Tchebycheff decomposition approach, is used in MOEA/D-UDM to alleviate the inconsistency between the weight vector of subproblem and the direction of its optimal solution in the Tchebycheff decomposition approach. The proposed MOEA/D-UDM is compared with two state-of-the-art MOEAs, namely MOEA/D and UMOEA/D on a number of MAPs. Experimental results suggest that the proposed MOEA/D-UDM outperforms or performs similarly to the other compared algorithms in terms of hypervolume and inverted generational distance metrics on different types of problems. The effects of uniform weight vector initializing method and the modified Tchebycheff decomposition are also studied separately.

A branch and bound method for the solution of multiparametric mixed integer linear programming problems

In this paper, we present a novel algorithm for the solution of multiparametric mixed integer linear programming (mp-MILP) problems that exhibit uncertain objective function coefficients and uncertain entries in the right-hand side constraint vector. The algorithmic procedure employs a branch and bound strategy that involves the solution of a multiparametric linear programming sub-problem at leaf nodes and appropriate comparison procedures to update the tree. McCormick relaxation procedures are employed to overcome the presence of bilinear terms in the model. The algorithm generates an envelope of parametric profiles, containing the optimal solution of the mp-MILP problem. The parameter space is partitioned into polyhedral convex critical regions. Two examples are presented to illustrate the steps of the proposed algorithm.

A Tri-Level Model of Centralized Transmission and Decentralized Generation Expansion Planning for an Electricity Market—Part I

We develop a tri-level model of transmission and generation expansion planning in a deregulated power market environment. Due to long planning/construction lead times and concerns for network reliability, transmission expansion is considered in the top level as a centralized decision. In the second level, multiple decentralized GENCOs make their own capacity expansion decisions while anticipating a wholesale electricity market equilibrium in the third level. The collection of bi-level games in the lower two levels forms an equilibrium problem with equilibrium constraints (EPEC) that can be approached by either the diagonalization method (DM) or a complementarity problem (CP) reformulation. We propose a hybrid iterative solution algorithm that combines a CP reformulation of the tri-level problem and DM solutions of the EPEC sub-problem.

Roy–Steiner equations forπNscattering

In this talk, we present a coupled system of integral equations for the πN → πN (s-channel) and ππ → N̅N (t-channel) lowest partial waves, derived from Roy–Steiner equations for pion–nucleon scattering. After giving a brief overview of this system of equations, we present the solution of the t-channel sub-problem by means of Muskhelishvili–Omnès techniques, and solve the s-channel sub-problem after finding a set of phase shifts and subthreshold parameters which satisfy the Roy–Steiner equations.

Cost-Sensitive Support Vector Machine Using Randomized Dual Coordinate Descent Method for Big Class-Imbalanced Data Classification

Cost-sensitive support vector machine is one of the most popular tools to deal with class-imbalanced problem such as fault diagnosis. However, such data appear with a huge number of examples as well as features. Aiming at class-imbalanced problem on big data, a cost-sensitive support vector machine using randomized dual coordinate descent method (CSVM-RDCD) is proposed in this paper. The solution of concerned subproblem at each iteration is derived in closed form and the computational cost is decreased through the accelerating strategy and cheap computation. The four constrained conditions of CSVM-RDCD are derived. Experimental results illustrate that the proposed method increases recognition rates of positive class and reduces average misclassification costs on real big class-imbalanced data.

An Inexact Sequential Quadratic Optimization Algorithm for Nonlinear Optimization

We propose a sequential quadratic optimization method for solving nonlinear optimization problems with equality and inequality constraints. The novel feature of the algorithm is that, during each iteration, the primal-dual search direction is allowed to be an inexact solution of a given quadratic optimization subproblem. We present a set of generic, loose conditions that the search direction (i.e., inexact subproblem solution) must satisfy so that global convergence of the algorithm for solving the nonlinear problem is guaranteed. The algorithm can be viewed as a globally convergent inexact Newton-based method. The results of numerical experiments are provided to illustrate the reliability of the proposed numerical method.

Vehicle Routing and Scheduling Problems for Convenience Store Industry Considering Soft Time Windows

Vehicle routing problem with soft time windows and simultaneous pickups and deliveries (VRPSTWSPD), which illustrates a class of distribution and routing systems of convenience store industry, is introduced. The VRPSTWSPD is a novel variant, where it combines characteristics of two existing variants in the literature. A mathematical model of the VRPSTWSPD is first developed in the paper. A branch-and-price exact-based procedure with a two-stage subproblem solution algorithm for solving small-sized instances is then described, followed by an efficient genetic algorithm metaheuristic-based procedure taking into account intermediate infeasible search strategy, which is more adequate for larger instances. Both solution algorithms were successfully tested on benchmark instances. The exact solutions can be used as references to evaluate the performances of the proposed genetic algorithm.

On the ellipticity of coupled finite element and one-equation boundary element methods for boundary value problems

In this paper we extend some recent results on the stability of the Johnson–Nedelec coupling of finite and boundary element methods in the case of boundary value problems. In Of and Steinbach (Z Angew Math Mech 93:476–484, 2013), Sayas (SIAM J Numer Anal 47:3451–3463, 2009) and Steinbach (SIAM J Numer Anal 49:1521–1531, 2011), the case of a free-space transmission problem was considered, and sufficient and necessary conditions are stated which ensure the ellipticity of the bilinear form for the coupled problem. The proof was based on considering the energies which are related to both the interior and exterior problem. In the case of boundary value problems for either interior or exterior problems, additional estimates are required to bound the energy for the solutions of related subproblems. Moreover, several techniques for the stabilization of the coupled formulations are analysed. Applications involve boundary value problems with either hard or soft inclusions, exterior boundary value problems, and macro-element techniques.

An image reconstruction method from Fourier data with uncertainties on the spatial frequencies

In this paper the reconstruction of a two-dimensional image from a nonuniform sampling of its Fourier transform is considered, in the presence of uncertainties on the frequencies corresponding to the measured data. The problem therefore becomes a blind deconvolution, in which the unknowns are both the image to be reconstructed and the exact frequencies. The availability of information on the image and the frequencies allows to reformulate the problem as a constrained minimization of the least squares functional. A regularized solution of this optimization problem is achieved by early stopping an alternating minimization scheme. In particular, a gradient projection method is employed at each step to compute an inexact solution of the minimization subproblems. The resulting algorithm is applied on some numerical examples arising in a real-world astronomical application.

Pit optimisation and life of mine scheduling for a tenement in the Central African Copperbelt

Open pit optimisation and scheduling tasks are an important part of mining ventures that have attracted considerable attention in the mining industry in the past couple of decades. The process of determining the optimum pit significantly impacts the calculation of mine life, which involves huge capital investment. This paper addresses the application of a heuristic algorithm for optimising the pit size and then life of mine scheduling for a copper-cobalt deposit in the Central African copper belt. The resource model was generated using a traditional geostatistical algorithm, specifically the kriging method. The methodology used was to generate a series of optimised nested pits by artificially varying the commodity price and applying a graph closure algorithm. The mine scheduling was generated by combining the solutions of different sub-problems. The deposit is optimised to 9.1 Mt with an average copper grade of 2.03% and cobalt grade of 0.47%. The total NPV generated from the pit is around 495 million US dollars. The results of our proposed algorithm were compared with linear relaxation of the mine scheduling problem. They demonstrated that our proposed algorithm is 30 times more efficient than the linear relaxation problem with an optimality gap of 7.4%.

An effective matheuristic for the capacitated total quantity discount problem

New generation algorithms focus on hybrid miscellaneous of exact and heuristic methods. Combining meta-heuristics and exact methods based on mathematical models appears to be a very promising alternative in solving many combinatorial optimization problems. In this paper we introduce a matheuristic method exploiting the optimal solution of mixed integer linear subproblems to solve the complex supplier selection problem of a company that needs to select a subset of suppliers so to minimize purchasing costs while satisfying products demand. Suppliers offer products at given prices and apply discounts according to the total quantity purchased. The problem, known as Total Quantity Discount Problem (TQDP), is strongly NP-hard thus motivating the study of effective and efficient heuristic methods. The proposed solution method is strengthened by the introduction of an Integer Linear Programming (ILP) refinement approach. An extensive computational analysis on a set of benchmark instances available in the literature has shown how the method is efficient and extremely effective allowing us to improve the existing best known solution values.

Generalized separable parameter space techniques for fitting 1K‐5K serial compartment models

Kinetic modeling is widely used to analyze dynamic imaging data, estimating kinetic parameters that quantify functional or physiologic processes in vivo. Typical kinetic models give rise to nonlinear solution equations in multiple dimensions, presenting a complex fitting environment. This work generalizes previously described separable nonlinear least-squares techniques for fitting serial compartment models with up to three tissue compartments and five rate parameters. The approach maximally separates the linear and nonlinear aspects of the modeling equations, using a formulation modified from previous basis function methods to avoid a potential mathematical degeneracy. A fast and robust algorithm for solving the linear subproblem with full user-defined constraints is also presented. The generalized separable parameter space technique effectively reduces the dimensionality of the nonlinear fitting problem to one dimension for 2K-3K compartment models, and to two dimensions for 4K-5K models. Exhaustive search fits, which guarantee identification of the true global minimum fit, required approximately 10 ms for 2K-3K and 1.1 s for 4K-5K models, respectively. The technique is also amenable to fast gradient-descent iterative fitting algorithms, where the reduced dimensionality offers improved convergence properties. The objective function for the separable parameter space nonlinear subproblem was characterized and found to be generally well-behaved with a well-defined global minimum. Separable parameter space fits with the Levenberg-Marquardt algorithm required fewer iterations than comparable fits for conventional model formulations, averaging 1 and 7 ms for 2K-3K and 4K-5K models, respectively. Sensitivity to initial conditions was likewise reduced. The separable parameter space techniques described herein generalize previously described techniques to encompass 1K-5K compartment models, enable robust solution of the linear subproblem with full user-defined constraints, and are amenable to rapid and robust fitting using iterative gradient-descent type algorithms.

Inexact Bregman iteration with an application to Poisson data reconstruction

This work deals with the solution of image restoration problems by an iterative regularization method based on the Bregman iteration. Any iteration of this scheme requires the exact computation of the minimizer of a function. However, in some image reconstruction applications, it is either impossible or extremely expensive to obtain exact solutions of these subproblems. In this paper, we propose an inexact version of the iterative procedure, where the inexactness in the inner subproblem solution is controlled by a criterion that preserves the convergence of the Bregman iteration and its features in image restoration problems. In particular, the method allows us to obtain accurate reconstructions also when only an overestimation of the regularization parameter is known. The introduction of the inexactness in the iterative scheme allows us to address image reconstruction problems from data corrupted by Poisson noise, exploiting the recent advances about specialized algorithms for the numerical minimization of the generalized Kullback–Leibler divergence combined with a regularization term. The results of several numerical experiments enable us to evaluate the proposed scheme for image deblurring or denoising in the presence of Poisson noise.

Truss Structural Optimization Based on Globally Convergent Version of MMA

The subproblem of the globally convergent version of method of moving asymptotes (GCMMA) was deeply studied and realized by Lagrange dual method, sequential unconstrained minimization technique (SUMT) and conjugate gradient method. Based on the convex, separable and conservative properties of subproblem, a dual problem was built by using Lagrange dual method. The dual problem was transformed into unconstrained optimization problem using SUMT and solved by conjugate gradient method. Finally, the feasibility of solution of subproblem is demonstrated by truss structural optimization problem. Comparing with other optimization algorithms, GCMMA can converge to the global optimal solution.

A Novel Algorithm Based on Divide-and-Merge Strategy for Single Machine Scheduling

Many optimization problems in the industrial engineering world, particularly in manufacturing systems, are very complex in nature and difficult to solve by conventional optimization techniques. The use of formal approaches is frequently limited owing to the complexity of the models or algorithms. In this paper, a novel approach is proposed for large scale single machine weighted tardiness problems, which combine heuristic algorithm and divide-and-merge strategy. Given an instance of a scheduling problem, our method divides the problem into subproblems using a feasible solution as a basis; then ach subproblem is optimized and the solutions of subproblems are then merged to form a complete solution to the original problem. The aim of proposed method is to find good quality solution within a reasonable amount of computational time. We test our algorithm on big single machine total weighted tardiness problem, which is an NP-hard problem. The results show that the proposed approach have promising application to some traditional hard problems, especially to single machine weighted tardiness problem with big scale.

Rapid blockwise multi-resolution clustering of facial images for intelligent watermarking

Population-based evolutionary computation (EC) is widely used to optimize embedding parameters in intelligent watermarking systems. Candidate solutions generated with these techniques allow finding optimal embedding parameters of all blocks of a cover image. However, using EC techniques for full optimization of a stream of high-resolution grayscale face images is very costly. In this paper, a blockwise multi-resolution clustering (BMRC) framework is proposed to reduce this cost. During training phase, solutions obtained from multi-objective optimization of reference face images are stored in an associative memory. During generalization operations, embedding parameters of an input image are determined by searching for previously stored solutions of similar sub-problems in memory, thereby eliminating the need for full optimization for the whole face image. Solutions for sub-problems correspond to the most common embedding parameters for a cluster of similar blocks in the texture feature space. BMRC identifies candidate block clusters used for embedding watermark bits using the robustness score metric. It measures the texture complexity of image block clusters and can thereby handle watermarks of different lengths. The proposed framework implements a multi-hypothesis approach by storing the optimization solutions according to different clustering resolutions and selecting the optimal resolution at the end of the watermarking process. Experimental results on the PUT face image database show a significant reduction in complexity up to 95.5 % reduction in fitness evaluations compared with reference methods for a stream of 198 face images.

An adaptive nonmonotone trust-region method with curvilinear search for minimax problem

In this paper we propose an adaptive nonmonotone algorithm for minimax problem. Unlike traditional nonmonotone method, the nonmonotone technique applied to our method is based on the nonmonotone technique proposed by Zhang and Hager [H.C. Zhang, W.W. Hager, A nonmonotone line search technique and its application to unconstrained optimization, SIAM J. Optim. 14(4)(2004) 1043–1056] instead of that presented by Grippo et al. [L. Grippo, F. Lampariello, S. Lucidi, A nonmonotone line search technique for Newton’s method, SIAM J. Numer. Anal. 23(4)(1986) 707–716]. Meanwhile, by using adaptive technique, it can adaptively perform the nonmonotone trust-region step or nonmonotone curvilinear search step when the solution of subproblems is unacceptable. Global and superlinear convergences of the method are obtained under suitable conditions. Preliminary numerical results are reported to show the effectiveness of the proposed algorithm.