237 research outputs found
An SDP Approach For Solving Quadratic Fractional Programming Problems
This paper considers a fractional programming problem (P) which minimizes a
ratio of quadratic functions subject to a two-sided quadratic constraint. As is
well-known, the fractional objective function can be replaced by a parametric
family of quadratic functions, which makes (P) highly related to, but more
difficult than a single quadratic programming problem subject to a similar
constraint set. The task is to find the optimal parameter and then
look for the optimal solution if is attained. Contrasted with the
classical Dinkelbach method that iterates over the parameter, we propose a
suitable constraint qualification under which a new version of the S-lemma with
an equality can be proved so as to compute directly via an exact
SDP relaxation. When the constraint set of (P) is degenerated to become an
one-sided inequality, the same SDP approach can be applied to solve (P) {\it
without any condition}. We observe that the difference between a two-sided
problem and an one-sided problem lies in the fact that the S-lemma with an
equality does not have a natural Slater point to hold, which makes the former
essentially more difficult than the latter. This work does not, either, assume
the existence of a positive-definite linear combination of the quadratic terms
(also known as the dual Slater condition, or a positive-definite matrix
pencil), our result thus provides a novel extension to the so-called "hard
case" of the generalized trust region subproblem subject to the upper and the
lower level set of a quadratic function.Comment: 26 page
S-Lemma with Equality and Its Applications
Let and be two quadratic functions
having symmetric matrices and . The S-lemma with equality asks when the
unsolvability of the system implies the existence of a real
number such that . The
problem is much harder than the inequality version which asserts that, under
Slater condition, is unsolvable if and only if for some . In this paper, we
show that the S-lemma with equality does not hold only when the matrix has
exactly one negative eigenvalue and is a non-constant linear function
(). As an application, we can globally solve as well as the two-sided generalized trust region subproblem
without any condition. Moreover, the
convexity of the joint numerical range where is a (possibly non-convex) quadratic
function and are affine functions can be characterized
using the newly developed S-lemma with equality.Comment: 34 page
Immunizing Conic Quadratic Optimization Problems Against Implementation Errors
We show that the robust counterpart of a convex quadratic constraint with ellipsoidal implementation error is equivalent to a system of conic quadratic constraints. To prove this result we first derive a sharper result for the S-lemma in case the two matrices involved can be simultaneously diagonalized. This extension of the S-lemma may also be useful for other purposes. We extend the result to the case in which the uncertainty region is the intersection of two convex quadratic inequalities. The robust counterpart for this case is also equivalent to a system of conic quadratic constraints. Results for convex conic quadratic constraints with implementation error are also given. We conclude with showing how the theory developed can be applied in robust linear optimization with jointly uncertain parameters and implementation errors, in sequential robust quadratic programming, in Taguchiās robust approach, and in the adjustable robust counterpart.Conic Quadratic Program;hidden convexity;implementation error;robust optimization;simultaneous diagonalizability;S-lemma
(Global) Optimization: Historical notes and recent developments
Recent developments in (Global) Optimization are surveyed in this paper. We collected and commented quite a large number of recent references which, in our opinion, well represent the vivacity, deepness, and width of scope of current computational approaches and theoretical results about nonconvex optimization problems. Before the presentation of the recent developments, which are subdivided into two parts related to heuristic and exact approaches, respectively, we briefly sketch the origin of the discipline and observe what, from the initial attempts, survived, what was not considered at all as well as a few approaches which have been recently rediscovered, mostly in connection with machine learning
An SDP approach to multi-level crossing minimization
We present an approach based on semidefinite programs (SDP) to tackle the multi-level crossing minimization prob- lem. Thereby, we are given a layered graph (i.e., the graphĀ“s vertices are assigned to multiple parallel levels) and ask for an ordering of the nodes on their levels such that, when draw- ing the graph with straight lines, the resulting number of crossings is minimized. Solving this step is crucial in the probably most widely used graph drawing scheme, the so- called Sugiyama framework. The problem has received a lot of attention both in the field of heuristics and exact methods. For a long time, integer linear programming (ILP) approaches were the only exact algorithms applicable at least to small graphs. Recently, SDP formulations for the special case of two levels were proposed and dominated the ILP for dense instances. In this paper, we present a new SDP formulation for the general multi-level version that, for two-levels, is even stronger than the aforementioned specialized SDP. As a side- product, we also obtain an SDP-based heuristic which in practice always gives (near-)optimal solutions. We conduct a large set of experiments, both on random- ized and on real-world instances, and compare our approach to a state-of-the-art ILP-based branch-and-cut implementa- tion. The SDP clearly dominates for denser graphs, while the ILP approach is usually faster for sparse instances. However, even for such sparse graphs, the SDP solves more instances to optimality than the ILP. In fact, there is no single instance the ILP solved, which the SDP did not. Overall, our experi- ments reveal that for sparse graphs, one should usually try to find an optimal solution with the ILP first. If this approach does not solve the instance to optimality within reasonable time, the SDP still has a good chance to do so. Being able to solve larger real-world instances than reported before, we are also able to evaluate heuristics for this problem. In this paper we do so for the traditional barycenter-heuristic (showing that it leaves a large gap to the true optimum) and the state-of-the-art upward-planarization method (showing that it is usually close to the optimum)
- ā¦