L-convexity is a concept of discrete convexity for functions defined on the integer lattice points, and plays a central role in the framework of discrete convex analysis. In this paper, we review recent development of algorithms for L-convex function minimization. We first point out the close connection between discrete convex analysis and various research fields such as discrete optimization, auction theory, and computer vision by showing that algorithms proposed independently in these research fields can be regarded as minimization algorithms applied to specific L-convex functions. Therefore, we can provide a unified approach to analyze the algorithms appearing in various research fields through the concept of L-convex function. We then present the recent results on theoretical bounds of the number of iterations required by some minimization algorithms, where precise bounds are given in terms of distance between the initial solution and the minimizer found by the algorithms. From these results we see that the algorithms output the “nearest” minimizer to the initial solution, and that the trajectories of solutions generated by the algorithms are “shortest paths” from the initial solution to the found minimizer. Finally, we consider an application of the results to iterative auctions in auction theory. We point out that the essence of the iterative auctions proposed by Ausubel (2006) lies in L-convexity. We also present new iterative auctions by Murota–Shioura–Yang (2016), which are based on the understanding of existing iterative auctions from the viewpoint of discrete convex analysis.

Pairs trading strategy has a history of at least 30 years in the stock market and is one of the most common trading strategies used today due to its understandability. Recently, Yamamoto and Hibiki [13] studied optimal pairs trading strategy using a new approach under actual fund management conditions, such as transaction costs, discrete rebalance intervals, finite investment horizons and so on. However, this approach cannot solve the problem of multiple pairs because this problem is formulated as a large scale simulation based non-continuous optimization problem. In this research, we formulate a model to solve an optimal pairs trading strategy problem using multiple pairs under actual fund management conditions. Furthermore, we propose a heuristic algorithm based on a derivative free optimization (DFO) method for solving this problem efficiently.

In constrained nonlinear optimization, the squared slack variables can be used to transform a problem with inequality constraints into a problem containing only equality constraints. This reformulation is usually not considered in the modern literature, mainly because of possible numerical instabilities. However, this argument only concerns the development of algorithms, and nothing stops us in using the strategy to understand the theory behind these optimization problems. In this note, we clarify the relation between the Karush-Kuhn-Tucker points of the original and the reformulated problems. In particular, we stress that the second-order sufficient condition is the key to establish their equivalence.

This paper discusses the error estimation of the last-column-block-augmented northwest-corner truncation (LC-block-augmented truncation, for short) of block-structured Markov chains (BSMCs) in continuous time. We first derive upper bounds for the absolute difference between the time-averaged functionals of a BSMC and its LC-block-augmented truncation, under the assumption that the BSMC satisfies the general f-modulated drift condition. We then establish computable bounds for a special case where the BSMC is exponentially ergodic. To derive such computable bounds for the general case, we propose a method that reduces BSMCs to be exponentially ergodic. We also apply the obtained bounds to level-dependent quasi-birth-and-death processes (LD-QBDs), and discuss the properties of the bounds through the numerical results on an M/M/s retrial queue, which is a representative example of LD-QBDs. Finally, we present computable perturbation bounds for the stationary distribution vectors of BSMCs.

This paper proposes a method for eliminating multicollinearity from linear regression models. Specifically, we select the best subset of explanatory variables subject to the upper bound on the condition number of the correlation matrix of selected variables. We first develop a cutting plane algorithm that, to approximate the condition number constraint, iteratively appends valid inequalities to the mixed integer quadratic optimization problem. We also devise a mixed integer semidefinite optimization formulation for best subset selection under the condition number constraint. Computational results demonstrate that our cutting plane algorithm frequently provides solutions of better quality than those obtained using local search algorithms for subset selection. Additionally, subset selection by means of our optimization formulation succeeds when the number of candidate explanatory variables is small.

The Gerber-Shiu function provides a way of measuring the risk of an insurance company. It is given by the expected value of a function that depends on the ruin time, the deficit at ruin, and the surplus prior to ruin. Its computation requires the evaluation of the overshoot/undershoot distributions of the surplus process at ruin. In this paper, we use the recent developments of the fluctuation theory and approximate it in a closed form by fitting the underlying process by phase-type Lévy processes. A sequence of numerical results are given.

This paper deals with a two-person zero-sum (TPZS) attrition game on a network in which attackers depart from a start node and attempt to reach a destination node while defenders deploy to intercept the attackers. Both players incur some attrition due to conflict between them, but the payoff of the game is the number of surviving attackers reaching the destination. We generate a system of models categorized according to various scenarios of the player's information acquisition (IA) about his opponent and derive linear programming formulations for the equilibria of the models. Comparing the equilibria, we evaluate the values of the situations around the IA in a comprehensive manner.

The authors analyse a search game involving an immobile hider and a searcher in which play takes place in discrete time on a network comprising a cycle together with a node 0 adjacent to a specified set of nodes in the cycle. The hider chooses a node of the cycle. Unaware of the hider's choice, the searcher starts at the node 0 and, at each subsequent time instant, moves from the node he occupies to an adjacent node and decides whether to search it. Play terminates when the searcher is at the node chosen by the hider and searches there. The searcher incurs a cost of one in moving from a node to an adjacent one and a search cost depending on whether or not the node is in the specified set. The searcher wants to minimize the costs of finding the hider and the hider to maximize them. We obtain upper bounds for the value of this game for the cases when the specified set has no adjacent nodes and when it is an interval. We show that these upper bounds are the value of the game in a number of cases.

Theorems of alternatives are useful mathematical tools in optimization. Their objects are pairs of linear systems. Folding a paper with a crease defines a linear inequality in \mathbb{R}². So one gets convex polygons by folding a paper many times. This paper provides a new perspective of duality to origami mathematics. We show that Gale's theorem of alternatives is useful for the study of twist fold.

Different strategies can be used to find a straight cable buried underground. The original problem considered by Faber et al. focused on a telephone company that wishes to dig a trench to locate a straight cable. The cable is known to pass within a given distance, a, from a marker erected above the putative location of the cable. Faber et al. showed that the shortest simply connected curve guaranteed to find the cable is a U-shaped curve whose length is about 18% less than that of a circular trench of radius a. This problem can be regarded as minimizing the maximum length that a trench digger must dig. In reality, once the cable is found, digging can stop. So far, however, no attempt has been made to evaluate the trench shape on characteristics other than the maximum trench length. In this paper, we present geometric probability models to analytically derive the distribution of trench length and calculate the expected value and variance for both the short-length (U-shaped) trench and a circular trench. Our main result is that the expected digging length is about 5% less for the circular trench than for the U-shaped trench.