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A bilevel program is an optimization problem whose constraints involve another optimization problem. This paper studies bilevel polynomial programs (BPPs), i.e., all the functions are polynomials. We reformulate BPPs equivalently as semi-infinite polynomial programs (SIPPs), using Fritz John conditions and Jacobian representations. Combining the exchange technique and Lasserre type semidefinite relaxations, we propose numerical methods for solving both simple and general BPPs. For simple BPPs, we prove the convergence to global optimal solutions. Numerical experiments are presented to show the efficiency of proposed algorithms.
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The bilevel program is an optimization problem where the constraint involves solutions to a parametric optimization problem. It is well-known that the value function reformulation provides an equivalent single-level optimization problem but it results in a nonsmooth optimization problem which never satisfies the usual constraint qualification such as the Mangasarian-Fromovitz constraint qualification (MFCQ). In this paper we show that even the first order sufficient condition for metric subregularity (which is in general weaker than MFCQ) fails at each feasible point of the bilevel program. We introduce the concept of directional calmness condition and show that under {the} directional calmness condition, the directional necessary optimality condition holds. {While the directional optimality condition is in general sharper than the non-directional one,} the directional calmness condition is in general weaker than the classical calmness condition and hence is more likely to hold. {We perform the directional sensitivity analysis of the value function and} propose the directional quasi-normality as a sufficient condition for the directional calmness. An example is given to show that the directional quasi-normality condition may hold for the bilevel program.
Low rank matrix recovery problems appear widely in statistics, combinatorics, and imaging. One celebrated method for solving these problems is to formulate and solve a semidefinite program (SDP). It is often known that the exact solution to the SDP with perfect data recovers the solution to the original low rank matrix recovery problem. It is more challenging to show that an approximate solution to the SDP formulated with noisy problem data acceptably solves the original problem; arguments are usually ad hoc for each problem setting, and can be complex. In this note, we identify a set of conditions that we call simplicity that limit the error due to noisy problem data or incomplete convergence. In this sense, simple SDPs are robust: simple SDPs can be (approximately) solved efficiently at scale; and the resulting approximate solutions, even with noisy data, can be trusted. Moreover, we show that simplicity holds generically, and also for many structured low rank matrix recovery problems, including the stochastic block model, $mathbb{Z}_2$ synchronization, and matrix completion. Formally, we call an SDP simple if it has a surjective constraint map, admits a unique primal and dual solution pair, and satisfies strong duality and strict complementarity. However, simplicity is not a panacea: we show the Burer-Monteiro formulation of the SDP may have spurious second-order critical points, even for a simple SDP with a rank 1 solution.
This paper studies bilevel polynomial optimization problems. To solve them, we give a method based on polynomial optimization relaxations. Each relaxation is obtained from the Kurash-Kuhn-Tucker (KKT) conditions for the lower level optimization and the exchange technique for semi-infinite programming. For KKT conditions, Lagrange multipliers are represented as polynomial or rational functions. The Moment-SOS relaxations are used to solve the polynomial optimizattion relaxations. Under some general assumptions, we prove the convergence of the algorithm for solving bilevel polynomial optimization problems. Numerical experiments are presented to show the efficiency of the method.
Chordal and factor-width decomposition methods for semidefinite programming and polynomial optimization have recently enabled the analysis and control of large-scale linear systems and medium-scale nonlinear systems. Chordal decomposition exploits the sparsity of semidefinite matrices in a semidefinite program (SDP), in order to formulate an equivalent SDP with smaller semidefinite constraints that can be solved more efficiently. Factor-width decompositions, instead, relax or strengthen SDPs with dense semidefinite matrices into more tractable problems, trading feasibility or optimality for lower computational complexity. This article reviews recent advances in large-scale semidefinite and polynomial optimization enabled by these two types of decomposition, highlighting connections and differences between them. We also demonstrate that chordal and factor-width decompositions allow for significant computational savings on a range of classical problems from control theory, and on more recent problems from machine learning. Finally, we outline possible directions for future research that have the potential to facilitate the efficient optimization-based study of increasingly complex large-scale dynamical systems.
Relaxed constant positive linear dependence constraint qualification (RCPLD) for a system of smooth equalities and inequalities is a constraint qualification that is weaker than the usual constraint qualifications such as Mangasarian Fromovitz constraint qualification and the linear constraint qualification. Moreover RCPLD is known to induce an error bound property. In this paper we extend RCPLD to a very general feasibility system which may include Lipschitz continuous inequality constraints, complementarity constraints and abstract constraints. We show that this RCPLD for the general system is a constraint qualification for the optimality condition in terms of limiting subdifferential and limiting normal cone and it is a sufficient condition for the error bound property under the strict complementarity condition for the complementarity system and Clarke regularity conditions for the inequality constraints and the abstract constraint set. Moreover we introduce and study some sufficient conditions for RCPLD including the relaxed constant rank constraint qualification (RCRCQ). Finally we apply our results to the bilevel program.
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