Nonconvex Terms Research Articles

A Vanishing Viscosity Approach to Quasistatic Evolution in Plasticity with Softening

We deal with quasistatic evolution problems in plasticity with softening, in the framework of small strain associative elastoplasticity. The presence of a nonconvex term due to the softening phenomenon requires a nontrivial extension of the variational framework for rate-independent problems to the case of a nonconvex energy functional. We argue that, in this case, the use of global minimizers in the corresponding incremental problems is not justified from the mechanical point of view. Thus, we analyse a different selection criterion for the solutions of the quasistatic evolution problem, based on a viscous approximation. This leads to a generalized formulation in terms of Young measures, developed in the first part of the paper. In the second part we apply our approach to some concrete examples.

Global Optimization of Reverse Osmosis Network for Wastewater Treatment and Minimization

Reverse osmosis (RO) has shown itself to be a viable technology for the treatment and minimization of industrial and domestic wastewater streams. The current research presents a deterministic branch-and-bound global optimization-based algorithm for the solution of the reverse osmosis network (RON) synthesis problem. The mathematical programming model describes the RON through nonconvex mixed-integer nonlinear programs (MINLPs). A piecewise mixed-integer linear program (MILP) is derived based on the convex relaxation of the nonconvex terms present in the MINLP formulation to approximate the original nonconvex program and to obtain a valid lower bound on the global optimum. The MILP model is solved at every node in the branch-and-bound tree to verify the global optimality of the treatment network within a pre-specified gap tolerance. Several constraints are developed to simultaneously screen the treatment network alternatives during the search, tighten the variable bounds, and consequently accelerate algorithm convergence. Water desalination is considered as a case study to illustrate the global optimization of the RO network.

Optimal Production Planning of Electricity from Coal-Fired Power Plant Networks under Environmental Considerations

In this study, electricity production from a network of coal power plants and under environmental constraints is considered. Several pollutants are incorporated into the decision planning process including nitrogen oxides (NOx), sulfur oxides (SOx) and mercury (Hg). A Nonconvex Mixed Integer Nonlinear Programming (MINLP) model was first developed to assist in the evaluation process. An exact linearization of the nonconvex terms was then derived to reformulate the MINLP model into a mixed integer linear program (MILP). Several scenarios of electricity production planning and emission reduction targets are analyzed on a representative case study.

Analytical bounds on the minimizers of (nonconvex) regularized least-squares

This is a theoretical study on the minimizers of cost-functions composed of an l 2 data-fidelity term and a possibly nonsmooth or nonconvex regularization term acting on the differences or the discrete gradients of the image or the signal to restore. More precisely, we derive general nonasymptotic analytical bounds characterizing the local and the global minimizers of these cost-functions. We first derive bounds that compare the restored data with the noisy data. For edge-preserving regularization, we exhibit a tight data-independent bound on the l ∞ norm of the residual (the estimate of the noise), even if its l 2 norm is being minimized. Then we focus on the smoothing incurred by the (local) minimizers in terms of the differences or the discrete gradient of the restored image (or signal).

Efficient Reconstruction of Piecewise Constant Images Using Nonsmooth Nonconvex Minimization

We consider the restoration of piecewise constant images where the number of the regions and their values are not fixed in advance, with a good difference of piecewise constant values between neighboring regions, from noisy data obtained at the output of a linear operator (e.g., a blurring kernel or a Radon transform). Thus we also address the generic problem of unsupervised segmentation in the context of linear inverse problems. The segmentation and the restoration tasks are solved jointly by minimizing an objective function (an energy) composed of a quadratic data-fidelity term and a nonsmooth nonconvex regularization term. The pertinence of such an energy is ensured by the analytical properties of its minimizers. However, its practical interest used to be limited by the difficulty of the computational stage which requires a nonsmooth nonconvex minimization. Indeed, the existing methods are unsatisfactory since they (implicitly or explicitly) involve a smooth approximation of the regularization term and often get stuck in shallow local minima. The goal of this paper is to design a method that efficiently handles the nonsmooth nonconvex minimization. More precisely, we propose a continuation method where one tracks the minimizers along a sequence of approximate nonsmooth energies $\{J_\eps\}$, the first of which being strictly convex and the last one the original energy to minimize. Knowing the importance of the nonsmoothness of the regularization term for the segmentation task, each $J_\eps$ is nonsmooth and is expressed as the sum of an $\ell_1$ regularization term and a smooth nonconvex function. Furthermore, the local minimization of each $J_{\eps}$ is reformulated as the minimization of a smooth function subject to a set of linear constraints. The latter problem is solved by the modified primal-dual interior point method, which guarantees the descent direction at each step. Experimental results are presented and show the effectiveness and the efficiency of the proposed method. Comparison with simulated annealing methods further shows the advantage of our method.

Optimal design of reverse-osmosis networks for wastewater treatment

An optimization study of reverse-osmosis networks (RON) for wastewater treatment has been carried out by describing the system as a nonconvex mixed-integer nonlinear problem (MINLP). A mixed-integer linear problem (MILP) is derived from the original nonlinear problem by the convex relaxation of the nonconvex terms in the MINLP to provide bounds for the global optimum. The MILP model is solved iteratively to supply different initial guesses for the nonconvex MINLP model. It is found that such a procedure is effective in finding local optimum solutions in reasonable time and overcoming possible convergence difficulties associated with MINLP local search methods. Examples of water desalination and wastewater treatment from the pulp and paper industry are considered as case studies to illustrate the proposed solution strategy.

An improved piecewise outer-approximation algorithm for the global optimization of MINLP models involving concave and bilinear terms

In this paper a new version of the Outer Approximation for Global Optimization Algorithm by Bergamini et al. [Bergamini, M.L., Aguirre, P., & Grossmann, I.E. (2005a). Logic based outer approximation for global optimization of synthesis of process networks. Computers and Chemical Engineering 29, 1914] is proposed, in order to speed up the convergence in nonconvex MINLP models that involve bilinear and concave terms. Bounding problems are constructed replacing these nonconvex terms by piecewise linear underestimators. These problems, which correspond to mixed-integer linear programs, are solved to generate approximate solutions with improved objective value. When no further feasible solution can be found, this guarantees that the upper bound cannot be improved in the nonconvex problem, thus providing a termination criterion. The new algorithm is applied to five different synthesis problems in the areas of water networks, heat exchanger networks and distillation sequences. The results show a significant reduction in the computational cost compared with the previous version of the algorithm.

Global Optimal Structures of Heat Exchanger Networks by Piecewise Relaxation

A new methodology for the global optimization of heat exchanger networks is presented, based on an outer approximation methodology, aided by physical insights. The problem is formulated as a mixed-integer nonlinear problem (MINLP). Two lower bounding convex MINLP problems are constructed, including piecewise underestimators of the nonconvex terms. A solution of the bounding problems gives an approximated optimal solution that is used as an initial point for solving the reduced NLP problem, or that proves there is no better solution. Meanwhile, the global bounding problem selects feasible structures with improved objective value. In order to reduce the number of feasible structures to be explored, rigorous constraints obtained from physical insights are included in the bounding problems. Networks with up to 9 process streams have been solved to global optimality, improving considerably the computing time required to solve them. Networks with more than 10 process streams have been solved with the global optimization strategy, giving as results a set of feasible structures, with total cost near the optimal value. This offers the possibility of alternatives to be analyzed, considering other aspects not reflected in the original MINLP model.

An optimization approach for solving signomial discrete programming problems with free variables

Signomial discrete programming (SDP) problems occur frequently in engineering design. This paper proposes a generalized method to solve SDP problems with free variables. An SDP problem with free variables is first converted into another one containing non-negative variables, and then various non-convex signomial terms are transformed such that the original SDP problem becomes a convex integer program solvable to obtain a globally optimal solution. Compared with current SDP methods, the proposed method is capable of dealing with free variables of an SDP problem and is guaranteed to converge to a global optimum. In addition, several computationally efficient convexification rules for signomial terms are presented to enhance the efficiency of the optimization approach. Numerical examples in real applications are presented to demonstrate the usefulness of the proposed method.

An Exact Reformulation Algorithm for Large Nonconvex NLPs Involving Bilinear Terms

Many nonconvex nonlinear programming (NLP) problems of practical interest involve bilinear terms and linear constraints, as well as, potentially, other convex and nonconvex terms and constraints. In such cases, it may be possible to augment the formulation with additional linear constraints (a subset of Reformulation-Linearization Technique constraints) which do not affect the feasible region of the original NLP but tighten that of its convex relaxation to the extent that some bilinear terms may be dropped from the problem formulation. We present an efficient graph-theoretical algorithm for effecting such exact reformulations of large, sparse NLPs. The global solution of the reformulated problem using spatial Branch-and Bound algorithms is usually significantly faster than that of the original NLP. We illustrate this point by applying our algorithm to a set of pooling and blending global optimization problems.

Improve-and-Branch Algorithm for the Global Optimization of Nonconvex NLP Problems

A new algorithm to solve nonconvex NLP problems is presented. It is based on the solution of two problems. The reformulated problem RP is a suitable reformulation of the original problem and involves convex terms and concave univariate terms. The main problem MP is a nonconvex NLP that outer-approximates the feasible region and underestimate the objective function. MP involves convex terms and terms which are the products of concave univariate functions and new variables. Fixing the variables in the concave terms, a convex NLP that overestimates the feasible region and underestimates the objective function is obtained from the MP. Like most of the deterministic global optimization algorithms, bounds on all the variables in the nonconvex terms must be provided. MP forces the objective value to improve and minimizes the difference of upper and lower bound of all the variables either to zero or to a positive value. In the first case, a feasible solution of the original problem is reached and the objective function is improved. In general terms, the second case corresponds to an infeasible solution of the original problem due to the existence of gaps in some variables. A branching procedure is performed in order to either prove that there is no better solution or reduce the domain, eliminating the local solution of MP that was found. The MP solution indicates a key point to do the branching. A bound reduction technique is implemented to accelerate the convergence speed. Computational results demonstrate that the algorithm compares very favorably to other approaches when applied to test problems and process design problems. It is typically faster and it produces very accurate results.

Global optimization for the synthesis of integrated water systems in chemical processes

In this paper, we address the problem of optimal synthesis of an integrated water system, where water using processes and water treatment operations are combined into a single network such that the total cost of obtaining freshwater for use in the water using operations, and treating wastewater is globally minimized. A superstructure that incorporates all feasible design alternatives for water treatment, reuse and recycle, is proposed. We formulate the optimization of this structure as a non-convex Non-Linear Programming (NLP) problem, which is solved to global optimality. The problem takes the form of a non-convex Generalized Disjunctive Program (GDP) if there is a flexibility of choosing different treatment technologies for the removal of the various contaminants in the wastewater streams. A new deterministic spatial branch and contract algorithm is proposed for optimizing such systems, in which piecewise under- and over-estimators are used to approximate the non-convex terms in the original model to obtain a convex relaxation whose solution gives a lower bound on the global optimum. These lower bounds are made to converge to the solution within a branch and bound procedure. Several examples are presented to illustrate the optimization of the integrated networks using the proposed algorithm.

Convergence and stability of a three-step iterative algorithm for a general quasi-variational inequality problem

We consider a general quasi-variational inequality problem involving nonlinear, nonconvex and nondifferentiable term in uniformly smooth Banach space. Using retraction mapping and fixed point method, we study the existence of solution of general quasi-variational inequality problem and discuss the convergence analysis and stability of a three-step iterative algorithm for general quasi-variational inequality problem. The theorems presented in this paper generalize, improve, and unify many previously known results in the literature.

Validated Linear Relaxations and Preprocessing: Some Experiments

Based on work originating in the early 1970s, a number of recent global optimization algorithms have relied on replacing an original nonconvex nonlinear program by convex or linear relaxations. Such linear relaxations can be generated automatically through an automatic differentiation process. This process decomposes the objective and constraints (if any) into convex and nonconvex unary and binary operations. The convex operations can be approximated arbitrarily well by appending additional constraints, while the domain must somehow be subdivided (in an overall branch-and-bound process or in some other local process) to handle nonconvex constraints. In general, a problem can be hard if even a single nonconvex term appears. However, certain nonconvex terms lead to easier-to-solve problems than others. Recently, Neumaier, Lebbah, Michel, ourselves, and others have paved the way to utilizing such techniques in a validated context.In this paper, we present a symbolic preprocessing step that provides a measure...

Multi-shock structure of roll waves

The special class of periodic travelling waves which is known as roll waves is investigated for nonhomogeneous hyperbolic equations of gas dynamics type. In this Note these equations are applied to shallow water flows in inclined open channels, but the results obtained are more general and far-reaching. The necessary conditions for the existence of a roll wave are derived. It is shown that for a nonconvex pressure term, multi-shock configurations of roll waves of finite amplitude exist. A new type of periodic travelling wave, which corresponds to the slug flow regime in two-layer flows, is found. To cite this article: A. Boudlal, V.Yu. Liapidevskii, C. R. Mecanique 332 (2004).

A reduced dimension branch-and-bound algorithm for molecular design

This paper addresses the solution of mixed-integer nonlinear mathematical programs in which the number of linear constraints far exceed the number of nonlinear constraints, and with most variables participating in the nonconvex terms. Some computer-aided molecular design models have these features. A branch-and-bound (BB) algorithm is proposed that is specifically tailored to solving such problems. In a conventional BB algorithm, branching is performed on all the search variables that appear in the nonlinear terms. This translates to a large number of node traversals. To overcome this problem, we have proposed a new strategy for branching on a set of branching functions, which depend linearly on the search variables. The branching functions are determined from a special tree function (STF) representation of both the objective function and constraints. In order to construct the corresponding linear underestimators, we have developed the Sweep method. The proposed algorithm scales well with problem size. Specifically, as the problem size increases, the computational effort increases almost linearly. The computational efficiency of the algorithm is demonstrated by solving a molecular design problem to global optimality.

On the solution of mixed-integer nonlinear programming models for computer aided molecular design

This paper addresses the efficient solution of computer aided molecular design (CAMD) problems, which have been posed as mixed-integer nonlinear programming models. The models of interest are those in which the number of linear constraints far exceeds the number of nonlinear constraints, and with most variables participating in the nonconvex terms. As a result global optimization methods are needed. A branch-and-bound algorithm (BB) is proposed that is specifically tailored to solving such problems. In a conventional BB algorithm, branching is performed on all the search variables that appear in the nonlinear terms. This translates to a large number of node traversals. To overcome this problem, we have proposed a new strategy for branching on a set of linear branching functions, which depend linearly on the search variables. This leads to a significant reduction in the dimensionality of the search space. The construction of linear underestimators for a class of functions is also presented. The CAMD problem that is considered is the design of optimal solvents to be used as cleaning agents in lithographic printing.

A fuzzy c-means variant for the generation of fuzzy term sets

A fuzzy c-means (FCM) variant is proposed for the generation of fuzzy term sets with 1 2 overlap. The proposed variant differs from the original mainly in two areas. The first modification ensures that two end terms take the maximum and minimum domain values as their centers. The second modification prevents the generation of non-convex fuzzy terms that often occurs with the original algorithm. The optimal number of terms and the optimal shape of the membership function associated with each term are determined based on the mean squared error criterion. The exponential weight, m, used in the algorithm is found to greatly affect the shape of the membership function. The effect of data size used for the generation of fuzzy terms is also discussed. A generalized π-shaped function with a tunable parameter along with its complement is developed to fit all term sets generated by the FCM variant using various m values.

A fuzzy c-medians variant for the generation of fuzzy term sets

A fuzzy c-medians variant is proposed for the generation of fuzzy term sets with one half overlap. The proposed variant is modified from the original algorithm mainly in two areas. The first modification ensures that two end terms take the maximum and minimum domain values as their centers. The second modification prevents the generation of non-convex fuzzy terms that often are obtained by the original algorithm. The optimal number of terms and the optimal shape of membership function associated with each term are determined based on the mean squared error criterion. The exponential weight, m, used in the algorithm is found to greatly affect the shape of membership function. The effect of data size used for the generation of fuzzy terms is also discussed. A generalized s-shaped function with a tunable parameter along with its complement (i.e., a generalized z-shaped function with a tunable parameter) is developed to fit all term sets generated by the fuzzy c-medians variant using various m values. © 2002 John Wiley & Sons, Inc.

Global optimization for clusters of flexible molecules—solvent–solute interaction energy calculations

The identification of the global minimum energy configuration of molecular clusters has found many useful applications, including the calculation of solvent–solute interactions for phase equilibria prediction. Given the need to reliably identify the global minimum and other low-energy configurations, a fully deterministic global optimization algorithm is proposed. The potential energy is calculated using the OPLS force field [J. Am. Chem. Soc. 118 (1996) 11225] and the minimization problem is formulated using 3 N–6 independent internal coordinates where N is the total number of atoms. A branch-and-bound framework is used to solve the problem. Tight convex underestimators have been derived for the non-convex terms and bounds for the dependent variables are calculated using interval arithmetic. The algorithm generates a converging sequence of upper and lower bounds. The algorithm is illustrated through the calculation of the global minimum configuration of a butane molecule, and a butane-ethylamine pair.