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.     

DCA based algorithms for multiple sequence alignment (MSA)

In the last years many techniques in bioinformatics have been developed for the central and complex problem of optimally aligning biological sequences. In this paper we propose a new optimization approach based on DC (Difference of Convex functions) programming and DC Algorithm (DCA) for the multiple sequence alignment in its equivalent binary linear program, called “Maximum Weight Trace” problem. This problem is beforehand recast as a polyhedral DC program with the help of exact penalty techniques in DC programming. Our customized DCA, requiring solution of a few linear programs, is original because it converges after finitely many iterations to a binary solution while it works in a continuous domain. To scale-up large-scale (MSA), a constraint generation technique is introduced in DCA. Preliminary computational experiments on benchmark data show the efficiency of the proposed algorithm DCAMSA, which generally outperforms some standard algorithms.     

DC programming techniques for solving a class of nonlinear bilevel programs

We propose a method for finding a global solution of a class of nonlinear bilevel programs, in which the objective function in the first level is a DC function, and the second level consists of finding a Karush-Kuhn-Tucker point of a quadratic programming problem. This method is a combination of the local algorithm DCA in DC programming with a branch and bound scheme well known in discrete and global optimization. Computational results on a class of quadratic bilevel programs are reported.     

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 solving Linear Complementarity Problems by DC programming and DCA

In this paper, we consider four optimization models for solving the Linear Complementarity (LCP) Problems. They are all formulated as DC (Difference of Convex functions) programs for which the unified DC programming and DCA (DC Algorithms) are applied. The resulting DCA are simple: they consist of solving either successive linear programs, or successive convex quadratic programs, or simply the projection of points on \mathbbR₊²ⁿ\mathbb{R}_{+}^{2n}. Numerical experiments on several test problems illustrate the efficiency of the proposed approaches in terms of the quality of the obtained solutions, the speed of convergence, and so on. Moreover, the comparative results with Lemke algorithm, a well known method for the LCP, show that DCA outperforms the Lemke method.     

An efficient combined DCA and B&;B using DC/SDP relaxation for globally solving binary quadratic programs

This paper addresses a new continuous approach based on the DC (Difference of Convex functions) programming and DC algorithms (DCA) to Binary quadratic programs (BQP) which play a key role in combinatorial optimization. DCA is completely different from other avalaible methods and featured by generating a convergent finite sequence of feasible binary solutions (obtained by solving linear programs with the same constraint set) with decreasing objective values. DCA is quite simple and inexpensive to handle large-scale problems. In particular DCA is explicit, requiring only matrix-vector products for Unconstrained Binary quadratic programs (UBQP), and can then exploit sparsity in the large-scale setting. To check globality of solutions computed by DCA, we introduce its combination with a customized Branch-and-Bound scheme using DC/SDP relaxation. The combined algorithm allows checking globality of solutions computed by DCA and restarting it if necessary and consequently accelerates the B&B approach. Numerical results on several series test problems provided in OR-Library (Beasley in J Global Optim, 8:429–433, 1996), show the robustness and efficiency of our algorithm with respect to standard methods. In particular DCA provides ϵ-optimal solutions in almost all cases after only one restarting and the combined DCA-B&B-SDP always provides (ϵ−)optimal solutions.     

A DC programming approach for solving the symmetric Eigenvalue Complementarity Problem

In this paper, we investigate a DC (Difference of Convex functions) programming technique for solving large scale Eigenvalue Complementarity Problems (EiCP) with real symmetric matrices. Three equivalent formulations of EiCP are considered. We first reformulate them as DC programs and then use DCA (DC Algorithm) for their solution. Computational results show the robustness, efficiency, and high speed of the proposed algorithms.     

Exact penalty and error bounds in DC programming

In the present paper, we are concerned with conditions ensuring the exact penalty for nonconvex programming. Firstly, we consider problems with concave objective and constraints. Secondly, we establish various results on error bounds for systems of DC inequalities and exact penalty, with/without error bounds, in DC programming. They permit to recast several class of difficult nonconvex programs into suitable DC programs to be tackled by the efficient DCA.     

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.     

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 study of the difference-of-convex approach for solving linear programs with complementarity constraints

This paper studies the difference-of-convex (DC) penalty formulations and the associated difference-of-convex algorithm (DCA) for computing stationary solutions of linear programs with complementarity constraints (LPCCs). We focus on three such formulations and establish connections between their stationary solutions and those of the LPCC. Improvements of the DCA are proposed to remedy some drawbacks in a straightforward adaptation of the DCA to these formulations. Extensive numerical results, including comparisons with an existing nonlinear programming solver and the mixed-integer formulation, are presented to elucidate the effectiveness of the overall DC approach.     

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.     

A continuous approach for the concave cost supply problem via DC programming and DCA

The paper addresses an important but difficult class of concave cost supply management problems which consist in minimizing a separable increasing concave objective function subject to linear and disjunctive constraints. We first recast these problems into mixed zero-one nondifferentiable concave minimization over linear constraints problems and then apply exact penalty techniques to state equivalent nondifferentiable polyhedral DC (Difference of Convex functions) programs. A new deterministic approach based on DC programming and DCA (DC Algorithms) is investigated to solve the latter ones. Finally numerical simulations are reported which show the efficiency, the robustness and the globality of our approach.     

A continuous DC programming approach to the strategic supply chain design problem from qualified partner set

We present a new continuous approach based on the DC (difference of convex functions) programming and DC algorithms (DCA) to the problem of supply chain design at the strategic level when production of a new market opportunity has to be launched among a set of qualified partners. A well known formulation of this problem is the mixed integer linear program. In this paper, we reformulate this problem as a DC program by using an exact penalty technique. The proposed algorithm is a combination of DCA and Branch and Bound scheme. It works in a continuous domain but provides mixed integer solutions. Numerical simulations on many empirical data sets show the efficiency of our approach with respect to the standard Branch and Bound algorithm.     

DC programming and DCA for globally solving the value-at-risk

The value-at-risk is an important risk measure that has been used extensively in recent years in portfolio selection and in risk analysis. This problem, with its known bilevel linear program, is reformulated as a polyhedral DC program with the help of exact penalty techniques in DC programming and solved by DCA. To check globality of computed solutions, a global method combining the local algorithm DCA with a well adapted branch-and-bound algorithm is investigated. An illustrative example and numerical simulations are reported, which show the robustness, the globality and the efficiency of DCA.     

An efficient DC programming approach for portfolio decision with higher moments

Portfolio selection with higher moments is a NP-hard nonconvex polynomial optimization problem. In this paper, we propose an efficient local optimization approach based on DC (Difference of Convex functions) programming—called DCA (DC Algorithm)—that consists of solving the nonconvex program by a sequence of convex ones. DCA will construct, in each iteration, a suitable convex quadratic subproblem which can be easily solved by explicit method, due to the proposed special DC decomposition. Computational results show that DCA almost always converges to global optimal solutions while comparing with the global optimization methods (Gloptipoly, Branch-and-Bound) and it outperforms several standard local optimization algorithms.     

Long-Short Portfolio Optimization Under Cardinality Constraints by Difference of Convex Functions Algorithm

In the matter of Portfolio selection, we consider an extended version of the Mean-Absolute Deviation (MAD) model, which includes discrete asset choice constraints (threshold and cardinality constraints) and one is allowed to sell assets short if it leads to a better risk-return tradeoff. Cardinality constraints limit the number of assets in the optimal portfolio and threshold constraints limit the amount of capital to be invested in (or sold short from) each asset and prevent very small investments in (or short selling from) any asset. The problem is formulated as a mixed 0–1 programming problem, which is known to be NP-hard. Attempting to use DC (Difference of Convex functions) programming and DCA (DC Algorithms), an efficient approach in non-convex programming framework, we reformulate the problem in terms of a DC program, and investigate a DCA scheme to solve it. Some computational results carried out on benchmark data sets show that DCA has a better performance in comparison to the standard solver IBM CPLEX.     

Solving continuous min max problem for single period portfolio selection with discrete constraints by DCA

In this article, we consider the application of a continuous min–max model with cardinality constraints to worst-case portfolio selection with multiple scenarios of risk, where the return forecast of each asset belongs to an interval. The problem can be formulated as minimizing a convex function under mixed integer variables with additional complementarity constraints. We first prove that the complementarity constraints can be eliminated and then use Difference of Convex functions (DC) programming and DC Algorithm (DCA), an innovative approach in non-convex programming frameworks, to solve the resulting problem. We reformulate it as a DC program and then show how to apply DCA to solve it. Numerical experiments on several test problems are reported that demonstrate the accuracy of the proposed algorithm.     

Strong formulations for quadratic optimization with M-matrices and indicator variables

We propose a DC (Difference of two Convex functions) formulation approach for sparse optimization problems having a cardinality or rank constraint. With the largest-k norm, an exact DC representation of the cardinality constraint is provided. We then transform the cardinality-constrained problem into a penalty function form and derive exact penalty parameter values for some optimization problems, especially for quadratic minimization problems which often appear in practice. A DC Algorithm (DCA) is presented, where the dual step at each iteration can be efficiently carried out due to the accessible subgradient of the largest-k norm. Furthermore, we can solve each DCA subproblem in linear time via a soft thresholding operation if there are no additional constraints. The framework is extended to the rank-constrained problem as well as the cardinality- and the rank-minimization problems. Numerical experiments demonstrate the efficiency of the proposed DCA in comparison with existing methods which have other penalty terms.     

Accelerating the DC algorithm for smooth functions

We introduce two new algorithms to minimise smooth difference of convex (DC) functions that accelerate the convergence of the classical DC algorithm (DCA). We prove that the point computed by DCA can be used to define a descent direction for the objective function evaluated at this point. Our algorithms are based on a combination of DCA together with a line search step that uses this descent direction. Convergence of the algorithms is proved and the rate of convergence is analysed under the ?ojasiewicz property of the objective function. We apply our algorithms to a class of smooth DC programs arising in the study of biochemical reaction networks, where the objective function is real analytic and thus satisfies the ?ojasiewicz property. Numerical tests on various biochemical models clearly show that our algorithms outperform DCA, being on average more than four times faster in both computational time and the number of iterations. Numerical experiments show that the algorithms are globally convergent to a non-equilibrium steady state of various biochemical networks, with only chemically consistent restrictions on the network topology.