Solve Maximization Problems Research Articles

Max Is More than Min: Solving Maximization Problems with Heuristic Search

Most work in heuristic search considers problems where a low cost solution is preferred (MIN problems). In this paper, we investigate the complementary setting where a solution of high reward is preferred (MAX problems). Example MAX problems include finding the longest simple path in a graph, maximal coverage, and various constraint optimization problems. We examine several popular search algorithms for MIN problems — optimal, suboptimal, and bounded suboptimal - and discover the curious ways in which they misbehave on MAX problems. We propose modifications that preserve the original intentions behind the algorithms but allow them to solve MAX problems, and compare them theoretically and empirically. Interesting results include the failure of bidirectional search and a discovered close relationships between Dijkstra's algorithm, weighted A*, and depth-first search. This work demonstrates that MAX problems demand their own heuristic search algorithms, which are worthy objects of study in their own right.

Optimization Techniques to Obtain the Best Combination of Alloy Strength and Toughness*

Mathematical tools are proposed for the optimization of chemical compositions of alloys and technological parameters of their processing during the development of new steel grades under considerable uncertainty. By reason of uncertainty, statistical models for mechanical properties of new alloys are used in optimization problems. Particularly, CVaR is used to estimate the right tail of CVN (Charpy V-Notch impact on toughness) distribution. This implies the use of nonconcave (convex) objective functions in maximization problems. Mathematical methods are proposed for solving maximization problems.

Explicit Solutions to Optimization Problems on the Intersections of the Unit Ball of the $l_1 $ and $l_\infty $ Norms with a Hyperplane

In this paper we characterize the extreme points of intersections of the $l_1 $ and $l_\infty $ unit balls in $R^n $ with a hyperplane. Such a characterization enables one to solve maximization problems of convex functions over these sets by enumeration. It turns out that the number of extreme points is $O(n^2 )$ under the $l_1 $ norm, and $O(2^N )$ under the $l_\infty $ norm. In particular, solving corresponding optimization problems by enumeration is efficient only under the $l_1 $ norm but not under the $l_\infty $ norm. Still, we get explicit solutions for maximization problems of linear objectives on the intersection of the $l_\infty $ unit ball with a hyperplane with a computational effort of $O(n)$. Applications of the results for computing bounds on the coefficients of ergodicity of square, nonnegative, irreducible matrices are discussed.