An improved definition of proper efficiency for vector maximization with respect to cones

AbstractRecently Borwein has proposed a definition for extending Geoffrion's concept of proper efficiency to the vector maximization problem in which the domination cone S is any nontrivial, closed convex cone. However, when S is the non-negative orthant, solutions may exist which are proper according to Borwein's definition but improper by Geoffrion's definition. As a result, when S is the non-negative orthant, certain properties of proper efficiency as defined by Geoffrion do not hold under Borwein's definition. To rectify this situation, we propose a definition of proper efficiency for the case when S is a nontrivial, closed convex cone which coincides with Geoffrion's definition when S is the non-negative orthant. The proposed definition seems preferable to Borwein's for developing a theory of proper efficiency for the case when S is a nontrivial, closed convex cone

Triage of patients with venous and lymphatic diseases during the COVID-19 pandemic – The Venous and Lymphatic Triage and Acuity Scale (VELTAS):: A consensus document of the International Union of Phlebology (UIP), Australasian College of Phlebology (ACP), American Vein and Lymphatic Society (AVLS), American Venous Forum (AVF), European College of Phlebology (ECoP), European Venous Forum (EVF), Interventional Radiology Society of Australasia (IRSA), Latin American Venous Forum, Pan-American Society of Phlebology and Lymphology and the Venous Association of India (VAI)

The coronavirus disease 2019 (COVID-19) global pandemic has resulted in diversion of healthcare resources to the management of patients infected with SARS-CoV-2 virus. Elective interventions and surgical procedures in most countries have been postponed and operating room resources have been diverted to manage the pandemic. The Venous and Lymphatic Triage and Acuity Scale was developed to provide an international standard to rationalise and harmonise the management of patients with venous and lymphatic disorders or vascular anomalies. Triage urgency was determined based on clinical assessment of urgency with which a patient would require medical treatment or surgical intervention. Clinical conditions were classified into six categories of: (1) venous thromboembolism (VTE), (2) chronic venous disease, (3) vascular anomalies, (4) venous trauma, (5) venous compression and (6) lymphatic disease. Triage urgency was categorised into four groups and individual conditions were allocated to each class of triage. These included (1) medical emergencies (requiring immediate attendance), example massive pulmonary embolism; (2) urgent (to be seen as soon as possible), example deep vein thrombosis; (3) semiurgent (to be attended to within 30-90 days), example highly symptomatic chronic venous disease, and (4) discretionary/nonurgent- (to be seen within 6-12 months), example chronic lymphoedema. Venous and Lymphatic Triage and Acuity Scale aims to standardise the triage of patients with venous and lymphatic disease or vascular anomalies by providing an international consensus-based classification of clinical categories and triage urgency. The scale may be used during pandemics such as the current COVID-19 crisis but may also be used as a general framework to classify urgency of the listed conditions

Note---Finding Certain Weakly-Efficient Vertices in Multiple Objective Linear Fractional Programming

Recently Kornbluth and Steuer have developed a simplex-based algorithm for finding all weakly-efficient vertices of an augmented feasible region of a multiple objective linear fractional programming problem. As part of this algorithm, they presented a method for detecting certain weakly-efficient vertices called break points. In this note we show that the procedure used by Kornbluth and Steuer in this method for computing the numbers needed to find these break points may sometimes fail. We also propose a fail-safe method for computing these numbers and give some computational results with this method.programming: multicriteria, fractional

Fractional programming with convex quadratic forms and functions

This article is concerned with two global optimization problems (P1) and (P2). Each of these problems is a fractional programming problem involving the maximization of a ratio of a convex function to a convex function, where at least one of the convex functions is a quadratic form. First, the article presents and validates a number of theoretical properties of these problems. Included among these properties is the result that, under a mild assumption, any globally optimal solution for problem (P1) must belong to the boundary of its feasible region. Also among these properties is a result that shows that problem (P2) can be reformulated as a convex maximization problem. Second, the article presents for the first time an algorithm for globally solving problem (P2). The algorithm is a branch and bound algorithm in which the main computational effort involves solving a sequence of convex programming problems. Convergence properties of the algorithm are presented, and computational issues that arise in implementing the algorithm are discussed. Preliminary indications are that the algorithm can be expected to provide a practical approach for solving problem (P2), provided that the number of variables is not too large

Branch-and-reduce algorithm for convex programs with additional multiplicative constraints

This article presents a branch-and-reduce algorithm for globally solving for the first time a convex minimization problem (P) with p[greater-or-equal, slanted]1 additional multiplicative constraints. In each of these p additional constraints, the product of two convex functions is constrained to be less than or equal to a positive number. The algorithm works by globally solving a 2p-dimensional master problem (MP) equivalent to problem (P). During a typical stage k of the algorithm, a point is found that minimizes the objective function of problem (MP) over a nonconvex set Fk that contains the portion of the boundary of the feasible region of the problem where a global optimal solution lies. If this point is feasible in problem (MP), the algorithm terminates. Otherwise, the algorithm continues by branching and creating a new, reduced nonconvex set Fk+1 that is a strict subset of Fk. To implement the algorithm, all that is required is the ability to solve standard convex programming problems and to implement simple algebraic steps. Convergence properties of the algorithm are given, and results of some computational experiments are reported.Global optimization Multiplicative programming Product of convex functions Branch-and-reduce