Linear and Integer Programming vs Linear Integration and Counting: A Duality Viewpoint

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In Conversation with The prize is awarded annually to scholars who have made fundamental, sustained contributions to theory in operations research and the management sciences. Since , five papers by Lasserre have appeared in Mathematics of OR , and in , Lasserre was appointed as an associate editor for the journal.

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INFORMS reached out to Jean Lasserre in Fall to learn more about his recent awards, his predictions for the future of operations research OR , and his advice for students entering this important field. What follows is the first in a series of interviews that will showcase the thought-leaders in the field of operations research. How has winning this award impacted your professional life?

Indeed the list of its important applications is almost endless. How has it impacted the field of optimization? Parrilo [ 11 ] used sums of squares decompositions for testing copositivity of a matrix and for some control applications. This hierarchy has finite convergence generically and provides the first optimality conditions for polynomial optimization with nonconvex analogue properties of the celebrated Karush—Kuhn—Tucker KTT [ 4,6 ] conditions in convex optimization.

Concerning real-world scenarios, the impact is more nuanced. The Lasserre Hierarchy has played a critical role in helping solve nonconvex problems of relatively modest size e. Optimum Power Flow problems might be the first real-world large-scale scenario where this methodology can outperform other approaches. In this respect, an important theoretical challenge is to understand the power and limitation of convex relaxations, especially for hard combinatorial optimization problems. This challenge is now central in the theoretical computer science TCS community and has already made its way into computer science courses at some prestigious universities.

Interestingly, in addition to optimization, Fourier analysis and techniques from quantum computing may also be appropriate and are explained and discussed in such courses. OR departments should introduce similar courses; for instance, powerful positivity certificates from real algebraic geometry and their dual facet on the K-moment problem should be taught in graduate courses. Because even if their proof is nontrivial and requires sophisticated mathematics, these positivity certificates are i easily understood, ii simple to implement, and iii can be used in many applications across different area.

This does not happen frequently! Another challenging problem is the pure integer programming problem IP. Indeed, simple examples of knapsack problems one constraint and variables! Surprisingly, all basic ingredients of the simplex algorithm basis, dual vector, reduced cost are also hidden in a beautiful formula e. In my book [ 10 ] , I have tried to popularize this point of view on LP, IP, integration, and counting, but with no success at all! Students should not be afraid of learning basic ideas of real and functional analysis, integration and measure theory, algebraic geometry, lattice points, and tools like Laplace and Fenchel transforms, Cauchy residues, and so forth.

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Of course, large-scale problems are still a challenge, even though we can solve problems of much larger size than in the s. It would be interesting to evaluate the respective roles that algorithms and computer power played in this achievement in the past 50 years. Some important applications e. This, in turn, has boosted research and renewed interest in large-scale, first-order methods for structured convex problems. Other inverse problems also appear in optimal control and robotics e.

Solving large-scale, semidefinite programs is another challenge. The early expectations of the s—that powerful interior-point methods would solve such problems—have not materialized. In contrast to LP solvers, which can solve huge size problems, and despite some progress, the size limitation of problems that can be solved by the current semidefinite programming SDP solvers is much more severe than for LP solvers. This is a pity because, for example, the powerful family of semidefinite convex relaxations for solving polynomial optimization problems is then limited to problems of modest size unless sparsity can be considered.

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Each convex relaxation in the hierarchy is a semidefinite program that, in principle, can be solved efficiently in time polynomial in its input size. However, as its size increases in the hierarchy, the convex relaxation becomes increasingly more difficult to solve.

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Fortunately, finite convergence eventually takes place generically. It is important to have fun and enjoy … and it helps spur creativity! It also recognizes my more recent work on linking LP, IP, linear integration, and counting problems. The Laplace and Z-transform are seen as analogues of the Legendre-Fenchel transform in the max-plus algebra.

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One of its main goals is to develop or provide alternatives to the moment-SOS hierarchy, so as to help solve difficult nonconvex problems in important applications e. Such problems are viewed as examples of the Generalized Problem of Moments, and the list of important applications is endless! But, this is an opportunity to emphasize how grateful I am to CNRS for providing me with a unique research environment with almost total freedom. In academic research, the freedom to pursue your intellectual interests is the most important feature that an employer can provide.

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