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Full-Text Articles in Computer Engineering
Adding Constraints -- A (Seemingly Counterintuitive But) Useful Heuristic In Solving Difficult Problems, Olga Kosheleva, Martine Ceberio, Vladik Kreinovich
Adding Constraints -- A (Seemingly Counterintuitive But) Useful Heuristic In Solving Difficult Problems, Olga Kosheleva, Martine Ceberio, Vladik Kreinovich
Departmental Technical Reports (CS)
Intuitively, the more constraints we impose on a problem, the more difficult it is to solve it. However, in practice, difficult-to-solve problems sometimes get solved when we impose additional constraints and thus, make the problems seemingly more complex. In this methodological paper, we explain this seemingly counter-intuitive phenomenon, and we show that, dues to this explanation, additional constraints can serve as a useful heuristic in solving difficult problems.
Towards An Efficient Bisection Of Ellipsoids, Paden Portillo, Martine Ceberio, Vladik Kreinovich
Towards An Efficient Bisection Of Ellipsoids, Paden Portillo, Martine Ceberio, Vladik Kreinovich
Departmental Technical Reports (CS)
Constraints are often represented as ellipsoids. One of the main advantages of such constrains is that, in contrast to boxes, over which optimization of even quadratic functions is NP-hard, optimization of a quadratic function over an ellipsoid is feasible. Sometimes, the area described by constrains is too large, so it is reasonable to bisect this area (one or several times) and solve the optimization problem for all the sub-areas. Bisecting a box, we still get a box, but bisecting an ellipsoid, we do not get an ellipsoid. Usually, this problem is solved by enclosing the half-ellipsoid in a larger ellipsoid, …
Constraint-Related Reinterpretation Of Fundamental Physical Equations Can Serve As A Built-In Regularization, Vladik Kreinovich, Juan Ferret, Martine Ceberio
Constraint-Related Reinterpretation Of Fundamental Physical Equations Can Serve As A Built-In Regularization, Vladik Kreinovich, Juan Ferret, Martine Ceberio
Departmental Technical Reports (CS)
Many traditional physical problems are known to be ill-defined: a tiny change in the initial condition can lead to drastic changes in the resulting solutions. To solve this problem, practitioners regularize these problem, i.e., impose explicit constraints on possible solutions (e.g., constraints on the squares of gradients). Applying the Lagrange multiplier techniques to the corresponding constrained optimization problems is equivalent to adding terms proportional to squares of gradients to the corresponding optimized functionals. It turns out that many optimized functionals of fundamental physics already have such squares-of-gradients terms. We therefore propose to re-interpret these equations -- by claiming that they …
Why Ellipsoid Constraints, Ellipsoid Clusters, And Riemannian Space-Time: Dvoretzky's Theorem Revisited, Karen Villaverde, Olga Kosheleva, Martine Ceberio
Why Ellipsoid Constraints, Ellipsoid Clusters, And Riemannian Space-Time: Dvoretzky's Theorem Revisited, Karen Villaverde, Olga Kosheleva, Martine Ceberio
Departmental Technical Reports (CS)
In many practical applications, we encounter ellipsoid constraints, ellipsoid-shaped clusters, etc. A usual justification for this ellipsoid shape comes from the fact that many real-life quantities are normally distributed, and for a multi-variate normal distribution, a natural confidence set (containing the vast majority of the objects) is an ellipsoid. However, ellipsoid appear more frequently than normal distributions (which occur in about half of the cases). In this paper, we provide a new justification for ellipsoids based on a known mathematical result -- Dvoretzky's Theorem.