Proportional symbol maps are a cartographic tool to assist in the visualization and analysis of quantitative data associated with specific locations, such as earthquake magnitudes, oil well production, and temperature at weather stations. As the name suggests, symbol sizes are proportional to the magnitude of the physical quantities that they represent. We present two novel … Read more
We propose an approach to the timing of markdowns over a finite time horizon that does not require the precise knowledge of the underlying probabilities, instead relying on range forecasts for the arrival rates of the demand processes, and that captures the degree of the manager’s risk aversion through intuitive budget-of-uncertainty functions. These budget functions … Read more
The question as to whether the Gomory-Chvatal closure of a non-rational polytope is a polytope has been a longstanding open problem in integer programming. In this paper, we answer this question in the affirmative, by combining ideas from polyhedral theory and the geometry of numbers. Article Download View The Gomory-Chvatal closure of a non-rational polytope … Read more
Minimization with orthogonality constraints (e.g., $X^\top X = I$) and/or spherical constraints (e.g., $\|x\|_2 = 1$) has wide applications in polynomial optimization, combinatorial optimization, eigenvalue problems, sparse PCA, p-harmonic flows, 1-bit compressive sensing, matrix rank minimization, etc. These problems are difficult because the constraints are not only non-convex but numerically expensive to preserve during iterations. … Read more
We propose a convex optimization formulation with the nuclear norm and $\ell_1$-norm to find a large approximately rank-one submatrix of a given nonnegative matrix. We develop optimality conditions for the formulation and characterize the properties of the optimal solutions. We establish conditions under which the optimal solution of the convex formulation has a specific sparse … Read more
Grothendieck inequalities are fundamental inequalities which are frequently used in many areas of mathematics and computer science. They can be interpreted as upper bounds for the integrality gap between two optimization problems: A difficult semidefinite program with rank-1 constraint and its easy semidefinite relaxation where the rank constrained is dropped. For instance, the integrality gap … Read more
The Lasserre hierarchy of semidefinite programming approximations to convex polynomial optimization problems is known to converge finitely under some assumptions. [J.B. Lasserre. Convexity in semialgebraic geometry and polynomial optimization. SIAM J. Optim. 19, 1995-2014, 2009.] We give a new proof of the finite convergence property, that does not require the assumption that the Hessian of … Read more
In this paper, we show that the Chatal-Gomory closure of a compact convex set is a rational polytope. This resolves an open question discussed in Schrijver 1980 and generalizes the same result for the case of rational polytopes (Schrijver 1980), rational ellipsoids (Dey and Vielma 2010) and strictly convex sets (Dadush et. al 2010). In … Read more
We consider in this paper quadratic programming problems with cardinality and minimum threshold constraints which arise naturally in various real-world applications such as portfolio selection and subset selection in regression. We propose a new semidefinite program (SDP) approach for computing the “best” diagonal decomposition that gives the tightest continuous relaxation of the perspective reformulation. We … Read more
The bundle-level method and their certain variants are known to exhibit an optimal rate of convergence, i.e., ${\cal O}(1/\sqrt{t})$, and also excellent practical performance for solving general non-smooth convex programming (CP) problems. However, this rate of convergence is significantly worse than the optimal one for solving smooth CP problems, i.e., ${\cal O}(1/t^2)$. In this paper, … Read more