準凸計画問題に対する劣微分を用いた最適性条件 (非線形解析学と凸解析学の研究)

本講究録では, 準凸計画問題に対する劣微分を用いた最適性条件について述べる. 特に近年筆者等によって示された, essentially quasiconvex programmingに対する最適性の必要十分条件, 一般の準凸計画問題に対する最適性の必要十分条件, 逆準凸制約を持つ準凸計画問題に対する最適性の必要条件について述べる

Two approaches toward constrained vector optimization and identity of the solutions

In this paper we deal with a Fritz John type constrained vector optimization problem. In spite that there are many concepts of solutions for an unconstrained vector optimization problem, we show the possibility “to double” the number of concepts when a constrained problem is considered. In particular we introduce sense I and sense II isolated minimizers, properly efficient points, efficient points and weakly efficient points. As a motivation leading to these concepts we give some results concerning optimality conditions in constrained vector optimization and stability properties of isolated minimizers and properly efficient points. Our main investigation and results concern relations between sense I and sense II concepts. These relations are proved mostly under convexity type conditions. Key words: Constrained vector optimization, Optimality conditions, Stability, Type of solutions and their identity, Vector optimization and convexity type conditions.

Well-Behavior, Well-Posedness and Nonsmooth Analysis

AMS subject classification: 90C30, 90C33.We survey the relationships between well-posedness and well-behavior. The latter notion means that any critical sequence (xn) of a lower semicontinuous function f on a Banach space is minimizing. Here “critical” means that the remoteness of the subdifferential ∂f(xn) of f at xn (i.e. the distance of 0 to ∂f(xn)) converges to 0. The objective function f is not supposed to be convex or smooth and the subdifferential ∂ is not necessarily the usual Fenchel subdifferential. We are thus led to deal with conditions ensuring that a growth property of the subdifferential (or the derivative) of a function implies a growth property of the function itself. Both qualitative questions and quantitative results are considered

Inverse Optimization: Closed-form Solutions, Geometry and Goodness of fit

In classical inverse linear optimization, one assumes a given solution is a candidate to be optimal. Real data is imperfect and noisy, so there is no guarantee this assumption is satisfied. Inspired by regression, this paper presents a unified framework for cost function estimation in linear optimization comprising a general inverse optimization model and a corresponding goodness-of-fit metric. Although our inverse optimization model is nonconvex, we derive a closed-form solution and present the geometric intuition. Our goodness-of-fit metric, $\rho$, the coefficient of complementarity, has similar properties to $R^2$ from regression and is quasiconvex in the input data, leading to an intuitive geometric interpretation. While $\rho$ is computable in polynomial-time, we derive a lower bound that possesses the same properties, is tight for several important model variations, and is even easier to compute. We demonstrate the application of our framework for model estimation and evaluation in production planning and cancer therapy

準凸計画間題に対するKKT条件と制約想定 (非線形解析学と凸解析学の研究)

本講究録では，準凸計画問題に対するKKT条件と制約想定について述べる．特に近年筆者によって示された， essentially quasiconvex programmingに対するGP劣微分と生成集合を用いた最適性の必要十分条件，一般の準凸計画問題に対するM劣微分と生成集合を用いた最適性の必要十分条件について述べる．特に既存の研究との関連や証明のアイディア等について詳細に述べる

NP-hardness of Deciding Convexity of Quartic Polynomials and Related Problems

We show that unless P=NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1992 when N. Z. Shor asked for the complexity of deciding convexity for quartic polynomials. We also prove that deciding strict convexity, strong convexity, quasiconvexity, and pseudoconvexity of polynomials of even degree four or higher is strongly NP-hard. By contrast, we show that quasiconvexity and pseudoconvexity of odd degree polynomials can be decided in polynomial time.Comment: 20 page

Algebraic Relaxations and Hardness Results in Polynomial Optimization and Lyapunov Analysis

This thesis settles a number of questions related to computational complexity and algebraic, semidefinite programming based relaxations in optimization and control.Comment: PhD Thesis, MIT, September, 201