Mixed-Integer Constrained Optimization Based on Memetic Algorithm

Evolutionary algorithms (EAs) are population-based global search methods. They have been successfully applied tomany complex optimization problems. However, EAs are frequently incapable of finding a convergence solution indefault of local search mechanisms. Memetic Algorithms (MAs) are hybrid EAs that combine genetic operators withlocal search methods. With global exploration and local exploitation in search space, MAs are capable of obtainingmore high-quality solutions. On the other hand, mixed-integer hybrid differential evolution (MIHDE), as an EA-basedsearch algorithm, has been successfully applied to many mixed-integer optimization problems. In this paper, amemetic algorithm based on MIHDE is developed for solving mixed-integer optimization problems. However, most ofreal-world mixed-integer optimization problems frequently consist of equality and/or inequality constraints. In order toeffectively handle constraints, an evolutionary Lagrange method based on memetic algorithm is developed to solvethe mixed-integer constrained optimization problems. The proposed algorithm is implemented and tested on twobenchmark mixed-integer constrained optimization problems. Experimental results show that the proposed algorithmcan find better optimal solutions compared with some other search algorithms. Therefore, it implies that the proposedmemetic algorithm is a good approach to mixed-integer optimization problems

Mirror-Descent Methods in Mixed-Integer Convex Optimization

In this paper, we address the problem of minimizing a convex function f over a convex set, with the extra constraint that some variables must be integer. This problem, even when f is a piecewise linear function, is NP-hard. We study an algorithmic approach to this problem, postponing its hardness to the realization of an oracle. If this oracle can be realized in polynomial time, then the problem can be solved in polynomial time as well. For problems with two integer variables, we show that the oracle can be implemented efficiently, that is, in O(ln(B)) approximate minimizations of f over the continuous variables, where B is a known bound on the absolute value of the integer variables.Our algorithm can be adapted to find the second best point of a purely integer convex optimization problem in two dimensions, and more generally its k-th best point. This observation allows us to formulate a finite-time algorithm for mixed-integer convex optimization

A Solver for Multiobjective Mixed-Integer Convex and Nonconvex Optimization

This paper proposes a general framework for solving multiobjective nonconvex optimization problems, i.e., optimization problems in which multiple objective functions have to be optimized simultaneously. Thereby, the nonconvexity might come from the objective or constraint functions, or from integrality conditions for some of the variables. In particular, multiobjective mixed-integer convex and nonconvex optimization problems are covered and form the motivation of our studies. The presented algorithm is based on a branch-and-bound method in the pre-image space, a technique which was already successfully applied for continuous nonconvex multiobjective optimization. However, extending this method to the mixed-integer setting is not straightforward, in particular with regard to convergence results. More precisely, new branching rules and lower bounding procedures are needed to obtain an algorithm that is practically applicable and convergent for multiobjective mixed-integer optimization problems. Corresponding results are a main contribution of this paper. What is more, for improving the performance of this new branch-and-bound method we enhance it with two types of cuts in the image space which are based on ideas from multiobjective mixed-integer convex optimization. Those combine continuous convex relaxations with adaptive cuts for the convex hull of the mixed-integer image set, derived from supporting hyperplanes to the relaxed sets. Based on the above ingredients, the paper provides a new multiobjective mixed-integer solver for convex problems with a stopping criterion purely in the image space. What is more, for the first time a solver for multiobjective mixed-integer nonconvex optimization is presented. We provide the results of numerical tests for the new algorithm. Where possible, we compare it with existing procedures

A parametric integer programming algorithm for bilevel mixed integer programs

We consider discrete bilevel optimization problems where the follower solves an integer program with a fixed number of variables. Using recent results in parametric integer programming, we present polynomial time algorithms for pure and mixed integer bilevel problems. For the mixed integer case where the leader's variables are continuous, our algorithm also detects whether the infimum cost fails to be attained, a difficulty that has been identified but not directly addressed in the literature. In this case it yields a ``better than fully polynomial time'' approximation scheme with running time polynomial in the logarithm of the relative precision. For the pure integer case where the leader's variables are integer, and hence optimal solutions are guaranteed to exist, we present two algorithms which run in polynomial time when the total number of variables is fixed.Comment: 11 page

Partially distributed outer approximation

This paper presents a novel partially distributed outer approximation algorithm, named PaDOA, for solving a class of structured mixed integer convex programming problems to global optimality. The proposed scheme uses an iterative outer approximation method for coupled mixed integer optimization problems with separable convex objective functions, affine coupling constraints, and compact domain. PaDOA proceeds by alternating between solving large-scale structured mixed-integer linear programming problems and partially decoupled mixed-integer nonlinear programming subproblems that comprise much fewer integer variables. We establish conditions under which PaDOA converges to global minimizers after a finite number of iterations and verify these properties with an application to thermostatically controlled loads and to mixed-integer regression