The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems

The DC programming and its DC algorithm (DCA) address the problem of minimizing a function f=g−h (with g,h being lower semicontinuous proper convex functions on R ⁿ ) on the whole space. Based on local optimality conditions and DC duality, DCA was successfully applied to a lot of different and various nondifferentiable nonconvex optimization problems to which it quite often gave global solutions and proved to be more robust and more efficient than related standard methods, especially in the large scale setting. The computational efficiency of DCA suggests to us a deeper and more complete study on DC programming, using the special class of DC programs (when either g or h is polyhedral convex) called polyhedral DC programs. The DC duality is investigated in an easier way, which is more convenient to the study of optimality conditions. New practical results on local optimality are presented. We emphasize regularization techniques in DC programming in order to construct suitable equivalent DC programs to nondifferentiable nonconvex optimization problems and new significant questions which have to be answered. A deeper insight into DCA is introduced which really sheds new light on DCA and could partly explain its efficiency. Finally DC models of real world nonconvex optimization are reported.     

DC approach to weakly convex optimization and nonconvex quadratic optimization problems

Motivated by weakly convex optimization and quadratic optimization problems, we first show that there is no duality gap between a difference of convex (DC) program over DC constraints and its associated dual problem. We then provide certificates of global optimality for a class of nonconvex optimization problems. As an application, we derive characterizations of robust solutions for uncertain general nonconvex quadratic optimization problems over nonconvex quadratic constraints.     

Optimizing a multi-stage production/inventory system by DC programming based approaches

This paper deals with optimizing the cost of set up, transportation and inventory of a multi-stage production system in presence of bottleneck. The considered optimization model is a mixed integer nonlinear program. We propose two methods based on DC (Difference of Convex) programming and DCA (DC Algorithm)—an innovative approach in nonconvex programming framework. The mixed integer nonlinear problem is first reformulated as a DC program and then DCA is developed to solve the resulting problem. In order to globally solve the problem, we combine DCA with a Branch and Bound algorithm (BB-DCA). A convex minorant of the objective function is introduced. DCA is used to compute upper bounds while lower bounds are calculated from a convex relaxation problem. The numerical results compared with those of COUENNE (http://www.coin-or.org/download/binary/Couenne/), a solver for mixed integer nonconvex programming, show the rapidity and the ?-globality of DCA in almost cases, as well as the efficiency of the combined DCA-Branch and Bound algorithm. We also propose a simple heuristic algorithm which is proved by experimental results to be better than an existing heuristic in the literature for this problem.     

Monotonic optimization for sensor cover energy problem

We study the sensor cover energy problem (SCEP) in wireless communication—a difficult nonconvex problem with nonconvex constraints. A local approach based on DC programming called DCA was proposed by Astorino and Miglionico (Optim Lett 10(2):355–368, 2016) for solving this problem. In the present paper, we propose a global approach to (SCEP) based on the theory of monotonic optimization. By using an appropriate reformulation of (SCEP) we propose an algorithm for finding quickly a local optimal solution along with an efficient algorithm for computing a global optimal solution. Computational experiments are reported which demonstrate the practicability of the approach.     

DC programming approaches for discrete portfolio optimization under concave transaction costs

The paper investigates DC programming and DCA for both modeling discrete portfolio optimization under concave transaction costs as DC programs, and their solution. DC reformulations are established by using penalty techniques in DC programming. A suitable global optimization branch and bound technique is also developed where a DC relaxation technique is used for lower bounding. Numerical simulations are reported that show the efficiency of DCA and the globality of its computed solutions, compared to standard algorithms for nonconvex nonlinear integer programs.     

Portfolio selection under downside risk measures and cardinality constraints based on DC programming and DCA

In this paper, we consider the case of downside risk measures with cardinality and bounding constraints in portfolio selection. These constraints limit the amount of capital to be invested in each asset as well as the number of assets composing the portfolio. While the standard Markowitz’s model is a convex quadratic program, this new model is a NP-hard mixed integer quadratic program. Realizing the computational intractability for this class of problems, especially large-scale problems, we first reformulate it as a DC program with the help of exact penalty techniques in Difference of Convex functions (DC) programming and then solve it by DC Algorithms (DCA). To check globality of computed solutions, a global method combining the local algorithm DCA with a Branch-and-Bound algorithm is investigated. Numerical simulations show that DCA is an efficient and promising approach for the considered problem.      

Hidden Convex Minimization

A class of nonconvex minimization problems can be classified as hidden convex minimization problems. A nonconvex minimization problem is called a hidden convex minimization problem if there exists an equivalent transformation such that the equivalent transformation of it is a convex minimization problem. Sufficient conditions that are independent of transformations are derived in this paper for identifying such a class of seemingly nonconvex minimization problems that are equivalent to convex minimization problems. Thus, a global optimality can be achieved for this class of hidden convex optimization problems by using local search methods. The results presented in this paper extend the reach of convex minimization by identifying its equivalent with a nonconvex representation.     

New and efficient algorithms for transfer prices and inventory holding policies in two-enterprise supply chains

We consider a multi-period problem of fair transfer prices and inventory holding policies in two enterprise supply chains. This problem was formulated as a mixed integer non-linear program by Gjerdrum et al. (Eur J Oper Res 143:582–599, 2002). Existing global optimization methods to solve this problem are computationally expensive. We propose a continuous approach based on difference of convex functions (DC) programming and DC Algorithms (DCA) for solving this combinatorial optimization problem. The problem is first reformulated as a DC program via an exact penalty technique. Afterward, DCA, an efficient local approach in non-convex programming framework, is investigated to solve the resulting problem. For globally solving this problem, we investigate a combined DCA-Branch and Bound algorithm. DCA is applied to get lower bounds while upper bounds are computed from a relaxation problem. The numerical results on several test problems show that the proposed algorithms are efficient: DCA provides a good integer solution in a short CPU time although it works on a continuous domain, and the combined DCA-Branch and Bound finds an \(\epsilon \) -optimal solution for large-scale problems in a reasonable time.     

A new global optimization method for univariate constrained twice-differentiable NLP problems

In this paper, a new global optimization method is proposed for an optimization problem with twice-differentiable objective and constraint functions of a single variable. The method employs a difference of convex underestimator and a convex cut function, where the former is a continuous piecewise concave quadratic function, and the latter is a convex quadratic function. The main objectives of this research are to determine a quadratic concave underestimator that does not need an iterative local optimizer to determine the lower bounding value of the objective function and to determine a convex cut function that effectively detects infeasible regions for nonconvex constraints. The proposed method is proven to have a finite ε-convergence to locate the global optimum point. The numerical experiments indicate that the proposed method competes with another covering method, the index branch-and-bound algorithm, which uses the Lipschitz constant.     

A combined DCA: GA for constructing highly nonlinear balanced boolean functions in cryptography

Substitution boxes, aka S-boxes, are a key component of modern crypto-systems. Several studies and developments were carried out on the problem of building high-quality S-boxes in the last few years. Qualities of such boxes, such as nonlinearity and balance, steer the robustness of modern block ciphers. This work is concerned with the construction of highly nonlinear balanced Boolean functions. A deterministic optimization model which is the minimization of a polyhedral convex function on a convex polytope with 0–1 variables is introduced. A local deterministic optimization approach called DCA (Difference of Convex functions Algorithm) is investigated. For finding a good starting point of DCA we propose two versions of a combined DCA–GA (Genetic Algorithm) method. Numerical simulations prove that DCA is a promising approach for this problem. Moreover the combination of DCA–GA improves the efficiency of DCA and outperforms other standard approaches.     

Fuzzy clustering based on nonconvex optimisation approaches using difference of convex (DC) functions algorithms

We present a fast and robust nonconvex optimization approach for Fuzzy C-Means (FCM) clustering model. Our approach is based on DC (Difference of Convex functions) programming and DCA (DC Algorithms) that have been successfully applied in various fields of applied sciences, including Machine Learning. The FCM model is reformulated in the form of three equivalent DC programs for which different DCA schemes are investigated. For accelerating the DCA, an alternative FCM-DCA procedure is developed. Experimental results on several real world problems that include microarray data illustrate the effectiveness of the proposed algorithms and their superiority over the standard FCM algorithm, with respect to both running-time and accuracy of solutions.     

On generalized trade-off directions in nonconvex multiobjective optimization

Trade-off information related to Pareto optimal solutions is important in multiobjective optimization problems with conflicting objectives. Recently, the concept of trade-off directions has been introduced for convex problems. These trade-offs are characterized with the help of tangent cones. Generalized trade-off directions for nonconvex problems can be defined by replacing convex tangent cones with nonconvex contingent cones. Here we study how the convex concepts and results can be generalized into a nonconvex case. Giving up convexity naturally means that we need local instead of global analysis. Received: December 2000 / Accepted: October 2001?Published online February 14, 2002     

Visualizing data as objects by DC (difference of convex) optimization

In this paper we address the problem of visualizing in a bounded region a set of individuals, which has attached a dissimilarity measure and a statistical value, as convex objects. This problem, which extends the standard Multidimensional Scaling Analysis, is written as a global optimization problem whose objective is the difference of two convex functions (DC). Suitable DC decompositions allow us to use the Difference of Convex Algorithm (DCA) in a very efficient way. Our algorithmic approach is used to visualize two real-world datasets.     

Convex underestimation for posynomial functions of positive variables

The approximation of the convex envelope of nonconvex functions is an essential part in deterministic global optimization techniques (Floudas in Deterministic Global Optimization: Theory, Methods and Application, 2000). Current convex underestimation algorithms for multilinear terms, based on arithmetic intervals or recursive arithmetic intervals (Hamed in Calculation of bounds on variables and underestimating convex functions for nonconvex functions, 1991; Maranas and Floudas in J Global Optim 7:143–182, (1995); Ryoo and Sahinidis in J Global Optim 19:403–424, (2001)), introduce a large number of linear cuts. Meyer and Floudas (Trilinear monomials with positive or negative domains: Facets of convex and concave envelopes, pp. 327–352, (2003); J Global Optim 29:125–155, (2004)), introduced the complete set of explicit facets for the convex and concave envelopes of trilinear monomials with general bounds. This study proposes a novel method to underestimate posynomial functions of strictly positive variables.     

A reformulation framework for global optimization

In this paper, we present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are used to obtain the convex underestimators. For more general nonconvex functions two versions of the so-called αBB-underestimator, valid for twice-differentiable functions, are integrated in the actual reformulation framework. However, in contrast to what is done in branch-and-bound type algorithms, no direct branching is performed in the actual algorithm. Instead a piecewise convex reformulation is used to convexify the entire problem in an extended variable-space, and the reformulated problem is then solved by a convex MINLP solver. As the piecewise linear approximations are made finer, the solution to the convexified and overestimated problem will form a converging sequence towards a global optimal solution. The result is an easily-implementable algorithm for solving a very general class of optimization problems.     

Limit vector variational inequality problems via scalarization

This paper presents a global optimization approach for solving signomial geometric programming (SGP) problems. We employ an accelerated extended cutting plane (ECP) approach integrated with piecewise linear (PWL) approximations to solve the global optimization of SGP problems. In this approach, we separate the feasible regions determined by the constraints into convex and nonconvex ones in the logarithmic domain. In the nonconvex feasible regions, the corresponding constraint functions are converted into mixed integer linear constraints using PWL approximations, while the other constraints with convex feasible regions are handled by the ECP method. We also use pre-processed initial cuts and batched cuts to accelerate the proposed algorithm. Numerical results show that the proposed approach can solve the global optimization of SGP problems efficiently and effectively.     

Optimization based DC programming and DCA for hierarchical clustering

One of the most promising approaches for clustering is based on methods of mathematical programming. In this paper we propose new optimization methods based on DC (Difference of Convex functions) programming for hierarchical clustering. A bilevel hierarchical clustering model is considered with different optimization formulations. They are all nonconvex, nonsmooth optimization problems for which we investigate attractive DC optimization Algorithms called DCA. Numerical results on some artificial and real-world databases are reported. The results demonstrate that the proposed algorithms are more efficient than related existing methods.     

A Global Optimization Method,QBB, for Twice-Differentiable Nonconvex Optimization Problem

A global optimization method, QBB, for twice-differentiable NLPs (Non-Linear Programming) is developed to operate within a branch-and-bound framework and require the construction of a relaxed convex problem on the basis of the quadratic lower bounding functions for the generic nonconvex structures. Within an exhaustive simplicial division of the constrained region, the rigorous quadratic underestimation function is constructed for the generic nonconvex function structure by virtue of the maximal eigenvalue analysis of the interval Hessian matrix. Each valid lower bound of the NLP problem with the division progress is computed by the convex programming of the relaxed optimization problem obtained by preserving the convex or linear terms, replacing the concave term with linear convex envelope, underestimating the special terms and the generic terms by using their customized tight convex lower bounding functions or the valid quadratic lower bounding functions, respectively. The standard convergence properties of the QBB algorithm for nonconvex global optimization problems are guaranteed. The preliminary computation studies are presented in order to evaluate the algorithmic efficiency of the proposed QBB approach.     

Bundle-based descent method for nonsmooth multiobjective DC optimization with inequality constraints

Multiobjective DC optimization problems arise naturally, for example, in data classification and cluster analysis playing a crucial role in data mining. In this paper, we propose a new multiobjective double bundle method designed for nonsmooth multiobjective optimization problems having objective and constraint functions which can be presented as a difference of two convex (DC) functions. The method is of the descent type and it generalizes the ideas of the double bundle method for multiobjective and constrained problems. We utilize the special cutting plane model angled for the DC improvement function such that the convex and the concave behaviour of the function is captured. The method is proved to be finitely convergent to a weakly Pareto stationary point under mild assumptions. Finally, we consider some numerical experiments and compare the solutions produced by our method with the method designed for general nonconvex multiobjective problems. This is done in order to validate the usage of the method aimed specially for DC objectives instead of a general nonconvex method.     

A Branch and Bound Method via d.c. Optimization Algorithms and Ellipsoidal Technique for Box Constrained Nonconvex Quadratic Problems

In this paper we propose a new branch and bound algorithm using a rectangular partition and ellipsoidal technique for minimizing a nonconvex quadratic function with box constraints. The bounding procedures are investigated by d.c. (difference of convex functions) optimization algorithms, called DCA. This is based upon the fact that the application of the DCA to the problems of minimizing a quadratic form over an ellipsoid and/or over a box is efficient. Some details of computational aspects of the algorithm are reported. Finally, numerical experiments on a lot of test problems showing the efficiency of our algorithm are presented.