Dedicated to Hoang Tuy on the occasion of his seventieth birthday Abstract. This paper is devoted to a thorough study on convex analysis approach to d.c. (difierence of convex functions) programming and gives the State of the Art. Main results about d.c. duality, local and global opti- malities in d.c. programming are presented. These materials constitute the basis of the DCA (d.c. algorithms). Its convergence properties have been tackled in detail, especially in d.c. polyhedral programming where it has flnite convergence. Exact penalty, Lagrangian duality without gap, and regularization techniques have beeen studied to flnd appropriate d.c. de- compositions and to improve consequently the DCA. Finally we present the application of the DCA to solving a lot of important real-life d.c. pro- grams.

Convex analysis approach to d.c. programming: Theory, Algorithm and Applications

This work addresses the development of an efficient solution strategy for obtaining global optima of continuous, integer, and mixed-integer nonlinear programs. Towards this end, we develop novel relaxation schemes, range reduction tests, and branching strategies which we incorporate into the prototypical branch-and-bound algorithm. In the theoretical/algorithmic part of the paper, we begin by developing novel strategies for constructing linear relaxations of mixed-integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties. We then use Lagrangian/linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming. This theory leads to new range reduction schemes, including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch-and-bound tree. Finally, we incorporate these relaxation and reduction strategies in a branch-and-bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems. In the computational part of the paper, we describe our implementation discussing, wherever appropriate, the use of suitable data structures and associated algorithms. We present computational experience with benchmark separable concave quadratic programs, fractional 0–1 programs, and mixed-integer nonlinear programs from applications in synthesis of chemical processes, engineering design, just-in-time manufacturing, and molecular design.

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Global optimization of mixed-integer nonlinear programs: A theoretical and computational study

Linearly constrained indefinite quadratic problems play an important role in global optimization. In this paper we study d.c. theory and its local approach to such problems. The new algorithm, CDA, efficiently produces local optima and sometimes produces global optima. We also propose a decomposition branch and bound method for globally solving these problems. Finally many numerical simulations are reported.

Solving a Class of Linearly Constrained Indefinite QuadraticProblems by D.C. Algorithms

A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs

This paper introduces the global mixed-integer quadratic optimizer, GloMIQO, a numerical solver addressing mixed-integer quadratically-constrained quadratic programs to $${\varepsilon}$$ -global optimality. The algorithmic components are presented for: reformulating user input, detecting special structure including convexity and edge-concavity, generating tight convex relaxations, partitioning the search space, bounding the variables, and finding good feasible solutions. To demonstrate the capacity of GloMIQO, we extensively tested its performance on a test suite of 399 problems of diverse size and structure. The test cases are taken from process networks applications, computational geometry problems, GLOBALLib, MINLPLib, and the Bonmin test set. We compare the performance of GloMIQO with respect to four state-of-the-art global optimization solvers: BARON 10.1.2, Couenne 0.4, LindoGLOBAL 6.1.1.588, and SCIP 2.1.0.

GloMIQO: Global mixed-integer quadratic optimizer

The global minimization of a large-scale linearly constrained concave quadratic problem is considered. The concave quadratic part of the objective function is given in terms of the nonlinear variablesx ∈R

 n
 , while the linear part is in terms ofy ∈R
k. For large-scale problems we may havek much larger thann. The original problem is reduced to an equivalent separable problem by solving a multiple-cost-row linear program with 2n cost rows. The solution of one additional linear program gives an incumbent vertex which is a candidate for the global minimum, and also gives a bound on the relative error in the function value of this incumbent. Ana priori bound on this relative error is obtained, which is shown to be ≤ 0.25, in important cases. If the incumbent is not a satisfactory approximation to the global minimum, a guaranteede-approximate solution is obtained by solving a single liner zero–one mixed integer programming problem. This integer problem is formulated by a simple piecewise-linear underestimation of the separable problem.

Global minimization of large-scale constrained concave quadratic problems by separable programming

The global minimization of large-scale concave quadratic problems over a bounded polyhedral set using a parallel branch and bound approach is considered. The objective function consists of both a concave part (nonlinear variables) and a strictly linear part, which are coupled by the linear constraints. These large-scale problems are characterized by having the number of linear variables much greater than the number of nonlinear variables. A linear underestimating function to the concave part of the objective is easily constructed and minimized over the feasible domain to get both upper and lower bounds on the global minimum function value. At each minor iteration of the algorithm, the feasible domain is divided into subregions and linear underestimating problems over each subregion are solved in parallel. Branch and bound techniques can then be used to eliminate parts of the feasible domain from consideration and improve the upper and lower bounds. It is shown that the algorithm guarantees that a solution is obtained to within any specified tolerance in a finite number of steps. Computational results are presented for problems with 25 and 50 nonlinear variables and up to 400 linear variables. These results were obtained on a four processor CRAY2 using both sequential and parallel implementations of the algorithm. The average parallel solution time was approximately 15 seconds for problems with 400 linear variables and a relative tolerance of 0.001. For a relative tolerance of 0.1, the average computation time appears to increase only linearly with the number of linear variables.

A parallel algorithm for constrained concave quadratic global minimization

The equivalence of zero–one integer programming and a concave quadratic penalty function problem has been shown by Raghavachari, for a sufficiently large value of the penalty. A lower bound for this penalty is obtained here, which in specific cases cannot be reduced.

Penalty for zero–one integer equivalent problem

A method is presented for the construction of test problems for which the global minimum point is known. Given a bounded convex polyhedron inR

 n
 , and a selected vertex, a concave quadratic function is constructed which attains its global minimum at the selected vertex. In general, this function will also have many other local minima.

Global minimum test problem construction

Construction of problems with known global solutions is important for the computational testing of constrained global minimization algorithms. In this paper, it is shown how to construct a concave quadratic function which attains its global minimum at a specified vertex of a polytope inRn+k. The constructed function is strictly concave in the variablesx ∈Rn and is linear in the variablesy ∈Rk. The number of linear variablesk may be much larger thann, so that large-scale global minimization test problems can be constructed by the methods described here.

J. B. Rosen

Papers

Global minimization of large-scale constrained concave quadratic problems by separable programming

A parallel algorithm for constrained concave quadratic global minimization

Penalty for zero–one integer equivalent problem

Global minimum test problem construction

Construction of large-scale global minimum concave quadratic test problems