An optimization problem for continuous submodular functions

"Real continuous submodular functions, as a generalization of the corresponding discrete notion to the continuous domain, gained considerable attention recently. The analog notion for entropy functions requires additional properties: a real function defined on the non-negative orthant of $\R^n$ is entropy-like (EL) if it is submodular, takes zero at zero, non-decreasing, and has the Diminishing Returns property. Motivated by problems concerning the Shannon complexity of multipartite secret sharing, a special case of the following general optimization problem is considered: find the minimal cost of those EL functions which satisfy certain constraints. In our special case the cost of an EL function is the maximal value of the $n$ partial derivatives at zero. Another possibility could be the supremum of the function range. The constraints are specified by a smooth bounded surface $S$ cutting off a downward closed subset. An EL function is feasible if at the internal points of $S$ the left and right partial derivatives of the function differ by at least one. A general lower bound for the minimal cost is given in terms of the normals of the surface $S$. The bound is tight when $S$ is linear. In the two-dimensional case the same bound is tight for convex or concave $S$. It is shown that the optimal EL function is not necessarily unique. The paper concludes with several open problems."

Determination of vehicle requirements of AGV system based on discrete event simulation and response surface methodology

The determination of AGV vehicle requirements in a manufacturing system has a great impact on the system performance. This paper first defines the AGV vehicle requirement determination as a general optimization problem, and secondly develops a new AGV vehicle requirement determination method capable of effective solving the problem. This method features with the combination of discrete event simulation (DES), sensitivity analysis, fractional factorial design (FFD) and response surface methodology (RSM). Tests and comparisons with other simulation based methods have shown that the proposed method combining the simulation method with analytical method, can make full use of their respective advantages and overcome the defects of existing methods. It is more practical.

Geometry and complexity of path integrals in inhomogeneous CFTs

In this work we develop the path integral optimization in a class of inhomogeneous 2d CFTs constructed by putting an ordinary CFT on a space with a position dependent metric. After setting up and solving the general optimization problem, we study specific examples, including the Möbius, SSD and Rainbow deformed CFTs, and analyze path integral geometries and complexity for universal classes of states in these models. We find that metrics for optimal path integrals coincide with particular slices of AdS3 geometries, on which Einstein’s equations are equivalent to the condition for minimal path integral complexity. We also find that while leading divergences of path integral complexity remain unchanged, constant contributions are modified in a universal, position dependent manner. Moreover, we analyze entanglement entropies in inhomogeneous CFTs and show that they satisfy Hill’s equations, which can be used to extract the energy density consistent with the first law of entanglement. Our findings not only support comparisons between slices of bulk spacetimes and circuits of path integrations, but also demonstrate that path integral geometries and complexity serve as a powerful tool for understanding the interesting physics of inhomogeneous systems.

Fusing Multiple Strategies in Population-Based Optimization Algorithm

A multi-strategy based population optimization, referred to MSPO, is proposed in this paper. The algorithm is developed by hybridizing four different population-based algorithms, bare bone particle swarm optimization, quantum-behaved particle swarm optimization, differential evolution and opposition-based learning. It aims at enhancing the exploration and exploitation capability of population based algorithm for general optimization problem. These four options are randomly selected with equal probability during the search process. The proposed algorithm is validated against test functions and then compares its performance with those of particle swarm optimization and bare bone particle swarm optimization. Numerical results show that the performance is increased greatly both in solution quality and convergent speed.

Generalized Differentiable -Invex Functions and Their Applications in Optimization

The concept of -convex function and its generalizations is studied with differentiability assumption. Generalized differentiable -convexity and generalized differentiable -invexity are used to derive the existence of optimal solution of a general optimization problem.

Flow Shop Production Scheduling under Uncertainty within Infinite Intermediate Storage

As to current situation of research on production scheduling under uncertainties in enterprises, Flow Shop production scheduling model is established based on the theory of fuzzy programming, in which fuzzy processing time is considered and the duration time of intermediate is unlimited. The maximum membership function of mean value has been applied to solve the non-linear fuzzy scheduling model in order to convert the fuzzy optimization problem to the general optimization problem. Finally, a cooperative co-evolutionary particle swarm optimization algorithm based on catastrophe added to improve the diversity of the swarm (CCPSO) is adopted to solve Flow Shop Production Scheduling Under Uncertainty within Infinite Intermediate Storage and results are obtained effectively.

EMPIRICAL COPULAS FOR CDO TRANCHE PRICING USING RELATIVE ENTROPY

We discuss the general optimization problem of choosing a copula with minimum entropy relative to a specified copula and a computationally intensive procedure to solve its dual. These techniques are applied to constructing an empirical copula for CDO tranche pricing. The empirical copula is chosen to be as close as possible to the industry standard Gaussian copula while ensuring a close fit to market tranche quotes. We find that the empirical copula performs noticeably better than the base correlation approach in pricing non-standard tranches and that the market view of default dependence is influenced by maturity.

A Method for Architecting Product Platforms With an Application to Interplanetary Mission Design

Consider a group of products sharing common parts and assemblies. The products in question we call a product family, and the common elements, the platform. A platform architecture is the set of selection and configuration choices shared among multiple products. We present a method for designing product platforms that takes into consideration both the performance requirements as well as the cost of the product family. The definition of a product platform is formulated as a general optimization problem in which the advantages of designing a common base must be balanced against the constraints of the individual product variants and of the whole family. This optimization approach has been implemented as an interactive multi-team-based negotiation model for designing an interplanetary spacecraft platform and its supported family variants. The model is used to identify possible subsystems of each spacecraft that could be made common to all or some of the missions, and the impact of architecture decisions on the performance of the product family.

Topology Optimization With Multiple Constraints

Topology optimization is used for determining the best layout of structural components to achieve predetermined performance goals. The density method which uses material density of each finite element as the design variable is employed. Unlike the most common approach which uses the optimality criteria methods, the topology design problem is formulated as a general optimization problem and is solved by the mathematical programming method. One of the major advantages of this approach is its generality; thus it can solve various problems, e.g. multi-objective and multi-constraint problems. In this study, the structural weight is chosen as the objective function and structural responses such as the compliances, displacements, and the natural frequencies are treated as the constraints. The MSC/NASTRAN finite element code is employed for response analyses. One example with four different optimization formulations was used to demonstrate this approach.