Hierarchy Of Semidefinite Relaxations Research Articles

Information Design in Nonatomic Routing Games With Partial Participation: Computation and Properties

We consider a routing game among nonatomic agents where link latency functions are conditional on an uncertain state of the network. The agents have the same prior belief about the state, but only a fixed fraction receive private route recommendations or a common message, which are generated by a known randomization, referred to as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">private</i> or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">public signaling policy</i> , respectively. The remaining agents choose route according to Bayes Nash flow with respect to the prior. In this article, we develop a computational approach to solve the optimal information design problem, i.e., to minimize expected social latency over all public or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">obedient</i> private signaling policies. For a fixed flow induced by nonparticipating agents, the design of an optimal private signaling policy has been shown to be a generalized problem of moments for polynomial link latency functions, and to admit an atomic solution with a provable upper bound on the number of atoms. This implies that, for polynomial link latency functions, information design can be equivalently cast as a polynomial optimization problem. This, in turn, can be arbitrarily lower bounded by a known hierarchy of semidefinite relaxations. The first level of this hierarchy is shown to be exact for the basic two link case with affine latency functions. We also identify a class of private signaling policies over which the optimal social cost is nonincreasing with an increasing fraction of participating agents for parallel networks. This is in contrast to existing results where the cost of participating agents under a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fixed</i> signaling policy may increase with their increasing fraction.

The Saddle Point Problem of Polynomials

This paper studies the saddle point problem of polynomials. We give an algorithm for computing saddle points. It is based on solving Lasserre’s hierarchy of semidefinite relaxations. Under some genericity assumptions on defining polynomials, we show that: (i) if there exists a saddle point, our algorithm can get one by solving a finite hierarchy of Lasserre-type semidefinite relaxations; (ii) if there is no saddle point, our algorithm can detect its nonexistence.

A hierarchy of semidefinite relaxations for completely positive tensor optimization problems

In this paper, we study the completely positive (CP) tensor program, which is a linear optimization problem with the cone of CP tensors and some linear constraints. We reformulate it as a linear program over the cone of moments, then construct a hierarchy of semidefinite relaxations for solving it. We also discuss how to find a best CP approximation of a given tensor. Numerical experiments are presented to show the efficiency of the proposed methods.

Optimal control of linear PDEs using occupation measures and SDP relaxations

This paper addresses the problem of solving a class of optimal control problems (OCPs) with infinite-dimensional linear state constraints involving Riesz-spectral operators. Each instance within this class has time/control-dependent polynomial Lagrangian cost and control constraints described by polynomials. We first perform a state-mode discretization of the Riesz-spectral operator. Then we approximate the resulting finite-dimensional OCPs by using a previously known hierarchy of semidefinite relaxations. Under certain compactness assumptions, we provide a converging hierarchy of semidefinite programming relaxations whose optimal values yield lower bounds for the initial OCP. We illustrate our method by two numerical examples, involving a diffusion partial differential equation and a wave equation. We also report on the related experiments.

A semidefinite method for tensor complementarity problems

In this paper, we study how to compute all real solutions of the tensor complimentary problem, if there are finite many ones. We formulate the problem as a sequence of polynomial optimization problems. The solutions can be computed sequentially. Each of them can be obtained by solving Lasserre's hierarchy of semidefinite relaxations. A semidefinite algorithm is proposed and its convergence properties are discussed. Some numerical experiments are also presented.

Tensor maximal correlation problems

This paper studies the tensor maximal correlation problem, which aims at optimizing correlations between sets of variables in many statistical applications. We reformulate the problem as an equivalent polynomial optimization problem, by adding the first order optimality condition to the constraints, then construct a hierarchy of semidefinite relaxations for solving it. The global maximizers of the problem can be detected by solving a finite number of such semidefinite relaxations. Numerical experiments show the efficiency of the proposed method.

Occupation measure methods for modelling and analysis of biological hybrid systems

Mechanistic models in biology often involve numerous parameters about which we do not have direct experimental information. The traditional approach is to fit these parameters using extensive numerical simulations (e.g. by the Monte-Carlo method), and eventually revising the model if the predictions do not correspond to the actual measurements. In this work we propose a methodology for hybrid system model revision, when new types of functions are needed to capture time varying parameters. To this end, we formulate a hybrid optimal control problem with intermediate points as successive infinite-dimensional linear programs (LP) on occupation measures. Then, these infinite-dimensional LPs are solved using a hierarchy of semidefinite relaxations. The whole procedure is applied on a recent model for haemoglobin production in erythrocytes.

Tensor eigenvalue complementarity problems

This paper studies tensor eigenvalue complementarity problems. Basic properties of standard and complementarity tensor eigenvalues are discussed. We formulate tensor eigenvalue complementarity problems as constrained polynomial optimization. When one tensor is strictly copositive, the complementarity eigenvalues can be computed by solving polynomial optimization with normalization by strict copositivity. When no tensor is strictly copositive, we formulate the tensor eigenvalue complementarity problem equivalently as polynomial optimization by a randomization process. The complementarity eigenvalues can be computed sequentially. The formulated polynomial optimization can be solved by Lasserre’s hierarchy of semidefinite relaxations. We show that it has finite convergence for generic tensors. Numerical experiments are presented to show the efficiency of proposed methods.

Semi-definite relaxations for optimal control problems with oscillation and concentration effects

Converging hierarchies of finite-dimensional semi-definite relaxations have been proposed for state-constrained optimal control problems featuring oscillation phenomena, by relaxing controls as Young measures. These semi-definite relaxations were later on extended to optimal control problems depending linearly on the control input and typically featuring concentration phenomena, interpreting the control as a measure of time with a discrete singular component modeling discontinuities or jumps of the state trajectories. In this contribution, we use measures introduced originally by DiPerna and Majda in the partial differential equations literature to model simultaneously, and in a unified framework, possible oscillation and concentration effects of the optimal control policy. We show that hierarchies of semi-definite relaxations can also be constructed to deal numerically with nonconvex optimal control problems with polynomial vector field and semialgebraic state constraints.

Certifying convergence of Lasserre’s hierarchy via flat truncation

Consider the optimization problem of minimizing a polynomial function subject to polynomial constraints. A typical approach for solving it globally is applying Lasserre’s hierarchy of semidefinite relaxations, based on either Putinar’s or Schmüdgen’s Positivstellensatz. A practical question in applications is: how to certify its convergence and get minimizers? In this paper, we propose flat truncation as a certificate for this purpose. Assume the set of global minimizers is nonempty and finite. Our main results are: (1) Putinar type Lasserre’s hierarchy has finite convergence if and only if flat truncation holds, under some generic assumptions; the same conclusion holds for the Schmüdgen type one under weaker assumptions. (2) Flat truncation is asymptotically satisfied for Putinar type Lasserre’s hierarchy if the archimedean condition holds; the same conclusion holds for the Schmüdgen type one if the feasible set is compact. (3) We show that flat truncation can be used as a certificate to check exactness of standard SOS relaxations and Jacobian SDP relaxations.

An algorithm for semi-infinite polynomial optimization

We consider the semi-infinite optimization problem: $$f^*:=\min_{\mathbf{x}\in\mathbf{X}} \bigl\{f(\mathbf{x}):g(\mathbf{x},\mathbf{y}) \leq 0, \forall\mathbf{y}\in\mathbf {Y}_\mathbf{x}\bigr\},$$ where f,g are polynomials and X⊂ℝn as well as Yx⊂ℝp, x∈X, are compact basic semi-algebraic sets. To approximate f∗ we proceed in two steps. First, we use the “joint+marginal” approach of Lasserre (SIAM J. Optim. 20:1995–2022, 2010) to approximate from above the function x↦Φ(x)=sup {g(x,y):y∈Yx} by a polynomial Φd≥Φ, of degree at most 2d, with the strong property that Φd converges to Φ for the L1-norm, as d→∞ (and in particular, almost uniformly for some subsequence (dl), l∈ℕ). Therefore, to approximate f∗ one may wish to solve the polynomial optimization problem \(f^{0}_{d}=\min_{\mathbf{x}\in\mathbf{X}} \{f(\mathbf{x}): \varPhi _{d}(\mathbf{x})\leq0\}\) via a (by now standard) hierarchy of semidefinite relaxations, and for increasing values of d. In practice d is fixed, small, and one relaxes the constraint Φd≤0 to Φd(x)≤e with e>0, allowing us to change e dynamically. As d increases, the limit of the optimal value \(f^{\epsilon}_{d}\) is bounded above by f∗+e.

Min-max and robust polynomial optimization

We consider the robust (or min-max) optimization problem $$J^*:=\max_{\mathbf{y}\in{\Omega}}\min_{\mathbf{x}}\{f(\mathbf{x},\mathbf{y}): (\mathbf{x},\mathbf{y})\in\mathbf{\Delta}\}$$ where f is a polynomial and $${\mathbf{\Delta}\subset\mathbb{R}^n\times\mathbb{R}^p}$$ as well as $${{\Omega}\subset\mathbb{R}^p}$$ are compact basic semi-algebraic sets. We first provide a sequence of polynomial lower approximations $${(J_i)\subset\mathbb{R}[\mathbf{y}]}$$ of the optimal value function $${J(\mathbf{y}):=\min_\mathbf{x}\{f(\mathbf{x},\mathbf{y}): (\mathbf{x},\mathbf{y})\in \mathbf{\Delta}\}}$$ . The polynomial $${J_i\in\mathbb{R}[\mathbf{y}]}$$ is obtained from an optimal (or nearly optimal) solution of a semidefinite program, the ith in the "joint + marginal" hierarchy of semidefinite relaxations associated with the parametric optimization problem $${\mathbf{y}\mapsto J(\mathbf{y})}$$ , recently proposed in Lasserre (SIAM J Optim 20, 1995-2022, 2010). Then for fixed i, we consider the polynomial optimization problem $${J^*_i:=\max\nolimits_{\mathbf{y}}\{J_i(\mathbf{y}):\mathbf{y}\in{\Omega}\}}$$ and prove that $${\hat{J}^*_i(:=\displaystyle\max\nolimits_{\ell=1,\ldots,i}J^*_\ell)}$$ converges to J* as i ? ?. Finally, for fixed ? ? i, each $${J^*_\ell}$$ (and hence $${\hat{J}^*_i}$$ ) can be approximated by solving a hierarchy of semidefinite relaxations as already described in Lasserre (SIAM J Optim 11, 796---817, 2001; Moments, Positive Polynomials and Their Applications. Imperial College Press, London 2009).