Reverse Convex Research Articles

Optimization by level set methods vi:generalizations of surrogate type reverse convex duality

We give some extensions of our duality theorems of [9] on the optimization problem (P) α =inf h(G), where G is a subset of a locally convex space Fsuch that F\G and are convex, to tne case whien G is a subsei of an arbitrary set F and is an arbitrary functional. We give some applications to the case when G is embedded into a family of sets where X is a parameter set.

Reverse convex programming

Reverse convex programs generally have disconnected feasible regions. Basic solutions are defined and properties of the latter and of the convex hull of the feasible region are derived. Solution procedures are discussed and a cutting plane algorithm is developed.

Linear programs with an additional reverse convex constraint

A constraintg(x)⩾0 is said to be a reverse convex constraint if the functiong is continuous and strictly quasi-convex. The feasible regions for linear programs with an additional reverse convex constraint are generally non-convex and disconnected. It is shown that the convex hull of the feasible region is a convex polytope and, as a result, there is an optimal solution on an edge of the polytope defined by only the linear constraints. The only possible edges which can contain such an optimal solution are characterized in relation to the best feasible vertex of the polytope defined by only the linear constraints. This characterization then provides a finite algorithm for finding a globally optimal solution.

Optimization Problems Subject to a Budget Constraint with Economies of Scale

This paper describes a finite procedure for locating a global minimum of a problem with linear objective constraints except for one nonlinear constraint, which is of the “reverse convex” variety; that is, the direction of the inequality is the opposite of that requited for a convex constraint. Budget constraints in which the cost functions are subject to economies of scale are typically of this form. An illustrative example of the procedure is provided.

On convergence of a global search strategy for reverse convex problems

On the base of global optimality conditions for RCP we develop the global search strategy (GSS) and focus on the global convergence of GSS giving a variant of the proof.

XIX. Account of some British Coins found near High Wycombe, in Buckinghamshire, in 1827; by John Norris, Esq. in a Letter to Henry Ellis, Esq. F.R.S. Secretary

I beg to trouble you, as Secretary of the Society of Antiquaries (of which I have long been an unworthy member), with drawings of five Coins, which I conceive to be ancient British, and which have been lately found, very accidentally, near High Wycombe, Bucks. These five coins, with six others, were found by a miller's boy upon a place called Keep-hill, in the parish of Chipping Wycombe, about half a mile from the town. The boy was sent to the spot, which is very near the mill, to look after his master's sheep, and happening to take up a flint, to grub at a fresh mole track, two of the coins fell out of a tubular cavity in the flint during the operation, and upon examination, nine more were found within, the whole of which the boy brought to his master's house, where they remained some days, till the boy's father, hearing of the discovery, got possession of them and carried them about for sale. They are of gold, about the tenth of an inch thick, the obverse concave, the reverse convex; they weigh about 3½ pennyweights, and are of the size represented. Tubular flints, similar to the one in which these coins were found, are very common in this stoney soil, and certainly, if a person wished to conceal his treasure, he could hardly hit upon a less suspicious receptacle. There are remains of artificial ditches still visible above Keep-hill, and I think it may fairly be presumed to have been a British strong-hold.