# LP-type problem

It's been a few weeks since I did any big new Wikipedia-writing projects, but yesterday in my graduate algorithms class I had a lecture on LP-type problems (a type of optimization problem that includes both linear programs and some related problems such as finding the smallest circle that surrounds a set of points) and discovered that Wikipedia was missing any information on them. So today I added a new article on LP-type problems.

Hi David, what about this 'Largest Placement of One Convex Polygon Inside Another' (section 2)is that also an LP-type problem (and worthy of wikipedia-mention)? In the top right corner of page 3 you see that the space of possiblities is a convex polyhedron (the linear inequalities for this problem get presented there), but i'm not sure what the function to be minimized over this polyhdron is and what still counts as 'LP-type'.

Thanks, but I'm not sure I see where in section 2 that it shows that the problem considered there (finding a largest similar copy of one convex polygon that can be placed inside another one) is actually LP-type. They say something hinting that it might be when they write that the space of possible placements can be represented as a four-dimensional convex polytope with O(mn) facets, but then they solve the problem by listing all vertices of the polytope rather than say minimizing a convex objective function. The same paper you linked to also has a brief reference to using LP algorithms to compute the largest homothetic copy of one convex polygon inside another (that is, not allowing rotations) from Sharir & Toledo CGTA 1994, but when I looked at that one it was just doing low-dimensional linear programming itself, not any kind of generalization of linear programming.

The 'Largest Placement of One Convex Polygon Inside Another' just seems very similar to the smallest enclosing circle problem, which is LP-type. The smallest enclosing circle problem is also viewable as the largest placement of a convex polygon (= the convex hull of the points) inside a circle. So the only difference is that we replace the circle with another convex polygon and it's remarkable to me that then we don't have an LP-type problem anymore.

Without rotation, placing the largest copy of one polygon inside another is LP-type, and can actually be formulated as an LP; that's in the CGTA 1994 paper. But if you allow rotation, you get multiple local optima rather than a single global optimum: for instance if both the inner and outer polygons are slightly distorted regular n-gons, then there are n different locally optimal placements (the ones where the two n-gons line up with each other). That's a phenomenon that doesn't happen in linear programs and it suggests to me that the problem is probably not LP-type.

Alright thanks.

### Comments:

**ext_1039177**:

**2012-02-11T06:06:11Z**

Hi David, what about this 'Largest Placement of One Convex Polygon Inside Another' (section 2)is that also an LP-type problem (and worthy of wikipedia-mention)? In the top right corner of page 3 you see that the space of possiblities is a convex polyhedron (the linear inequalities for this problem get presented there), but i'm not sure what the function to be minimized over this polyhdron is and what still counts as 'LP-type'.

**11011110**:

**2012-02-11T07:27:43Z**

Thanks, but I'm not sure I see where in section 2 that it shows that the problem considered there (finding a largest similar copy of one convex polygon that can be placed inside another one) is actually LP-type. They say something hinting that it might be when they write that the space of possible placements can be represented as a four-dimensional convex polytope with O(mn) facets, but then they solve the problem by listing all vertices of the polytope rather than say minimizing a convex objective function. The same paper you linked to also has a brief reference to using LP algorithms to compute the largest homothetic copy of one convex polygon inside another (that is, not allowing rotations) from Sharir & Toledo CGTA 1994, but when I looked at that one it was just doing low-dimensional linear programming itself, not any kind of generalization of linear programming.

**ext_1039177**:

**2012-02-11T19:04:20Z**

The 'Largest Placement of One Convex Polygon Inside Another' just seems very similar to the smallest enclosing circle problem, which is LP-type. The smallest enclosing circle problem is also viewable as the largest placement of a convex polygon (= the convex hull of the points) inside a circle. So the only difference is that we replace the circle with another convex polygon and it's remarkable to me that then we don't have an LP-type problem anymore.

**11011110**:

**2012-02-11T19:11:01Z**

Without rotation, placing the largest copy of one polygon inside another is LP-type, and can actually be formulated as an LP; that's in the CGTA 1994 paper. But if you allow rotation, you get multiple local optima rather than a single global optimum: for instance if both the inner and outer polygons are slightly distorted regular n-gons, then there are n different locally optimal placements (the ones where the two n-gons line up with each other). That's a phenomenon that doesn't happen in linear programs and it suggests to me that the problem is probably not LP-type.

**ext_1039177**:

**2012-02-11T19:36:15Z**

Alright thanks.