Continuous Minmax Research Articles

Covariance Matrix Adaptation Evolutionary Strategy with Worst-Case Ranking Approximation for Min–Max Optimization and Its Application to Berthing Control Tasks

In this study, we consider a continuous min–max optimization problem minx∈ 𝕏maxy∈ 𝕐f(x, y) whose objective function is a black-box. We propose a novel approach to minimize the worst-case objective functionF(x) = maxy∈ 𝕐f(x, y) directly using a covariance matrix adaptation evolution strategy in which the rankings of solution candidates are approximated by our proposed worst-case ranking approximation mechanism. We develop two variants of worst-case ranking approximation combined with a covariance matrix adaptation evolution strategy and approximate gradient ascent as numerical solvers for the inner maximization problem. Numerical experiments show that our proposed approach outperforms several existing approaches when the objective function is a smooth strongly convex–concave function and the interaction betweenxandyis strong. We investigate the advantages of the proposed approach for problems where the objective function is not limited to smooth strongly convex–concave functions. The effectiveness of the proposed approach is demonstrated in the robust berthing control problem with uncertainty.

A proximal point algorithm for generalized fractional programs

In this paper, the problem of solving generalized fractional programs will be addressed. This problem has been extensively studied and several algorithms have been proposed. In this work, we propose an algorithm that combines the proximal point method with a continuous min–max formulation of discrete generalized fractional programs. The proposed method can handle non-differentiable convex problems with possibly unbounded feasible constraints set, and solves at each iteration a convex program with unique dual solution. It generates two sequences that approximate the optimal value of the considered problem from below and from above at each step. For a class of functions, including the linear case, the convergence rate is at least linear.

Efficient processing of continuous min–max distance bounded query with updates in road networks

In the last decade, the research community focuses on the design of various methods in support of spatio-temporal queries in road networks. In this paper, we present a novel and important query, named the continuous min–max distance bounded query (CM2DBQ). Given a moving query object q, a minimal distance dm, and a maximal distance dM, a CM2DBQ retrieves the bounded objects whose road distances to q are within the range [dm,dM] at each time instant. We propose two algorithms, named the Continuous Within Query-based (CWQ-based) algorithm and the CM2DBQalgorithm, to efficiently determine the bounded objects. In addition, we design a mechanism, named the bounded objects updating mechanism, to rapidly evaluate the new query result when object updates occur. Extensive experiments using real road network dataset demonstrate the efficiency of the proposed algorithms and the usefulness of the bounded objects updating mechanism.

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.

Modified Dinkelbach-Type Algorithm for Generalized Fractional Programs with Infinitely Many Ratios

In this paper, we extend the Dinkelbach-type algorithm of Crouzeix, Ferland, and Schaible to solve minmax fractional programs with infinitely many ratios. Parallel to the case with finitely many ratios, the task is to solve a sequence of continuous minmax problems, $$P(\alpha_{k})=\min_{x\in X}\left(\max_{t\in T}\left[f_{t}(x)-\alpha_{k}g_{t}(x) \right]\right)$$ , until {α k } converges to the root of P(α)=0. The solution of P(α k ) is used to generate αk+1. However, calculating the exact optimal solution of P(α k ) requires an extraordinary amount of work. To improve, we apply an entropic regularization method which allows us to solve each problem P(α k ) incompletely, generating an approximate sequence $$\{\tilde{\alpha}_{k}\}$$ , while retaining the linear convergence rate under mild assumptions. We present also numerical test results on the algorithm which indicate that the new algorithm is robust and promising.

Solving Continuous Min-Max Problems by an Iterative Entropic Regularization Method

We propose a method of outer approximations, with each approximate problem smoothed using entropic regularization, to solve continuous min-max problems. By using a well-known uniform error estimate for entropic regularization, convergence of the overall method is shown while allowing each smoothed problem to be solved inexactly. In the case of convex objective function and linear constraints, an interior-point algorithm is proposed to solve the smoothed problem inexactly. Numerical examples are presented to illustrate the behavior of the proposed method.

Solving spread spectrum radar polyphase code design problem by tabu search and variable neighbourhood search

A basic variable neighbourhood search (VNS) heuristic is applied for the first time to continuous min–max global optimization problems. The method is tested on a class of NP-hard global optimization problems arising from the synthesis of radar polyphase codes that has already been successfully treated by tabu search. The computational results show that VNS in average outperforms tabu search.

Polynomial Algorithms for a Class of Discrete Minmax Linear Programming Problems

We consider the problem of obtaining integer solutions to a minmax linear programming problem. Although this general problem is NP-complete, it is shown that a restricted version of this problem can be solved in polynomial time. For this restricted class of problems two polynomial time algorithms are suggested, one of which is strongly polynomial whenever its continuous analogue and an associated linear programming problem can be solved by a strongly polynomial algorithm. Our algorithms can also be used to obtain integer solutions for the minmax transportation problem with an inequality budget constraint. The equality constrained version of this problem is shown to be NP-complete. We also provide some new insights into the solution procedures for the continuous minmax linear programming problem.

Minmax linear knapsack problem with grouped variables and gub

Let be a partition of be a partition of Si and ci and aj are scalars for We consider the problem MinimiseMaximum Subject to An algorithm is proposed to solve this problem. We also obtain a linear tune algorithm to solve the knapsack sharing problem with piece-wise linear trade-off functions which improves its existing time bound. As a by-product of this result, we have an O(n) algorithm to solve the continuous minmax knapsack problem with GLB constraints which improves the existing time bound

Rival Models: Min-Max Problems and Algorithms

There exist rival models corresponding to rival theories of the economic system. Methods are discussed for using these models simultaneously in policy design. All models are assumed to be nonlinear. Rival models of the same system are utilised in worst-case design formulations. Wher there is a single model, subject to disturbances, a worst-case design approach is used to account for the effect of the disturbances. For disturbances in rival models, a robust control formulation is adopted in addition to the worst-case design problem for the rival models. Algorithms are discussed for solving the corresponding discrete and continuous min-max problems.

Linearization method for continuous min-max problem

Linearization method for continuous min-max problem