Lasserre's Hierarchy in Polynomial Optimization

• Lasserre's Hierarchy is a sequence of SDP relaxations for polynomial optimization that leverages moment and sums-of-squares duality to tighten lower bounds.
• It incorporates extensions for complex variables, multi-order relaxations, and symmetry-induced block-diagonal structures to enhance scalability.
• Its practical impact is evidenced by efficient solutions in large-scale optimal power flow problems with guaranteed convergence through flat extension conditions.

Lasserre's Hierarchy

Lasserre's hierarchy is a sequence of convex semidefinite programming (SDP) relaxations for polynomial optimization problems (POPs) which leverages the duality between moments and sums-of-squares (SOS) polynomials. It systematically tightens lower bounds on the global optimum of a POP by increasing the order of relaxation, and under suitable conditions, converges to the exact solution. Recent advances have generalized the framework to deal with complex variables, exploited problem sparsity via multi-ordered relaxations, introduced symmetry-driven block-diagonal structures, and demonstrated tractability on large-scale industrial applications such as optimal power flow (OPF) (Josz et al., 2017).

1. Core Construction: Classical Real-variable Lasserre Hierarchy

Given a basic semialgebraic set $$K = \{x \in \mathbb{R}^n \mid g_j(x) \ge 0, \, j=1, \ldots, m\}$$ and a polynomial $f \in \mathbb{R}[x]$, consider the polynomial optimization problem $p^* = \inf_{x \in K} f(x)$. Lasserre's hierarchy constructs a sequence of SDP relaxations parameterized by relaxation order $d$:

• Truncated moment sequence: Introduce variables $y = (y_\alpha)_{|\alpha| \le 2d}$ to represent approximate moments $\int x^\alpha \, d\mu$ for an unknown measure $\mu$ supported on $K$.
• Moment matrix: $M_d(y)$ is a symmetric matrix with entries $M_d(y)_{\alpha, \beta} = y_{\alpha+\beta}$, $|\alpha|,|\beta|\le d$.
• Localizing matrices: For each constraint $g_j$, of degree $d_j$, the localizing matrix is defined by $$M_{d-d_j}(g_jy)_{α, β} = \sum_\gamma (g_j)_\gamma y_{\alpha+\beta+\gamma}$$.

The $d$th order moment relaxation is the SDP: $$\min_{y} L_y(f) \quad \text{s.t.} \quad y_0 = 1,\, M_d(y) \succeq 0,\, M_{d-d_j}(g_jy)\succeq 0,\, j=1,\ldots,m$$ with $L_y(f) = \sum_\alpha f_\alpha y_\alpha$.

The dual SDP uses sums-of-squares multipliers: $$\max_{\lambda,\,\sigma_j} \lambda \quad \text{such that} \quad f(x) - \lambda = \sigma_0(x) + \sum_{j=1}^m \sigma_j(x)g_j(x)$$ where $\sigma_0$ is SOS of degree $\leq 2d$, and $\sigma_j$ is SOS of degree $\le 2(d - d_j)$.

Finite convergence at order $d$ is certified by the flat extension condition: $$\mathrm{rank} M_d(y^*) = \mathrm{rank} M_{d-1}(y^*)$$, allowing extraction of global minimizers (Josz et al., 2017).

2. Generalization to Complex Variables and Hermitian SOS

For applications in oscillatory physical systems (notably OPF), the Lasserre hierarchy is extended to complex variables $z \in \mathbb{C}^n$. Here, $f(z, \bar z)$ and $g_j(z, \bar z)$ are real-valued in $(z, \bar z)$; the feasible set is $$K = \{z \in \mathbb{C}^n \mid g_j(z, \bar z) \ge 0\}$$.

• Complex moment matrix: Now a Hermitian matrix indexed by double multi-indices $(\alpha,\beta)$ with $|\alpha|,|\beta|\le d$, entries $$M_d(y)_{(\alpha,\beta),(\gamma,\delta)}=y_{\alpha+\gamma,\beta+\delta}$$.
• Complex localizing matrices: $M_{d-k_j}(g_jy)_{(\alpha,\beta),(\gamma,\delta)}$ formed analogously for constraint $g_j$ of bidegree $k_j$.
• Hermitian SOS: The dual relaxation imposes

$$f(z, \bar z) - \lambda = \sigma_0(z,\bar z) + \sum_j \sigma_j(z, \bar z) g_j(z, \bar z)$$

with each $\sigma_j$ a Hermitian SOS: $\sigma(z,\bar z) = \sum_\ell |p_\ell(z)|^2$ for polynomials $p_\ell$ of degree $\le d - k_j$.

• Finite convergence: Requires both a flat extension

$$\mathrm{rank}\, M_d(y^*) = \mathrm{rank} M_{d - d_K}(y^*) \quad (d_K = \max\{2, k_1, ..., k_m\})$$

and verification of hyponormality of certain compressed shift operators, via $3\times 3$ block matrices built from $M_d(y)$ and moment matrices with variable multipliers (Josz et al., 2017).

3. Exploiting Sparsity: Multi-Order Lasserre Hierarchy

Many large-scale POPs exhibit substantial sparsity. The multi-ordered hierarchy assigns a potentially different relaxation order $d_j$ to each constraint $g_j$:

• Chordal sparsity decomposition: Build a variable interaction graph via monomial support, chordally complete it, and extract maximal cliques $C_1, ..., C_p$.
• Blockwise SOS representation: Formulate the SOS decomposition using variables restricted to clique supports:

$$f - \lambda = \sum_k \left[ \sigma_0^k(z_{C_k}, \bar z_{C_k}) + \sum_{g_j\in C_k} \sigma_j^k(z_{C_k}, \bar z_{C_k}) g_j \right]$$

• Mismatch-driven order selection: Iterative algorithm:
  1. Initialize all $d_j = 0$ or $1$.
  2. Solve moment SDP, build a candidate solution.
  3. Identify the $h$ most violated constraints and increment their orders.
  4. Repeat until all residues are below tolerance.

• Global convergence: As each $d_j \to \infty$, the hierarchy recovers the dense Lasserre relaxation and converges globally under standard Archimedean assumptions (Josz et al., 2017).

4. Symmetry-induced Block-diagonal Structure

If the objective and constraints are invariant under a compact group $G$, symmetry can be leveraged to decompose the (potentially very large) moment and localizing matrices into smaller blocks:

• Torus invariance ($z \mapsto e^{i\theta} z$): Moment entries $y_{\alpha, \beta}$ survive only for $|\alpha| = |\beta|$; blocks are indexed by total degree.
• Sign invariance ($x \mapsto -x$): Only even-degree moment entries survive; $M_d(y)$ splits into even and odd components.

This block-diagonalization reduces the order and size of SDPs by a factor roughly corresponding to the number of symmetry blocks, which is crucial for scalability in high-dimensional problems (Josz et al., 2017).

5. Practical Impact: Large-scale Optimal Power Flow and Beyond

The aforementioned extensions and innovations have enabled the Lasserre hierarchy to solve real-world polynomial optimization instances at unprecedented scale:

• OPF applications: The power flow equations with $n \approx 4500$ complex voltages and $\sim$14,500 constraints are optimally solved with moment relaxations.
  • Complex Hermitian SOS reduces block sizes by approximately $2^d$, producing identical relaxation bounds as real variable hierarchies but with greater tractability.
  • Sparsity and symmetry further reduce the moment block sizes: in large test cases, only a small fraction of constraints needed increased relaxation order (typically $d_j \le 2$).
  • Enforcing torus symmetry decomposed each moment block into up to 4 sub-blocks for $d=3$.
  • Sample times: Polish 3012-wp ($n \approx 4584$): first bounds in 320s (real) vs 141s (complex); global solution in 900s (real) vs 700s (complex).
  • All computed global minimizers satisfied problem constraints to within $0.005$~p.u. (voltage) and $1$~MVA (power) (Josz et al., 2017).

Deployment of these scalable, certified global methods extends to other domains involving nonconvex polynomial equations with structure: signal processing, control, large-scale combinatorial/graph-theoretic relaxations, and more.

6. Convergence, Duality, and Flat Extension

Theoretical completeness is maintained across all extensions:

• Under the Archimedean property (existence of a ball constraint), the hierarchy delivers a monotone sequence of lower bounds converging upwards to the global infimum (Tacchi, 2020).
• Strong duality holds between the primal moment SDP and the SOS dual at each order, provided boundedness (via ball constraint) (Josz et al., 2014).
• Finite convergence is certified by the flat extension property (Curto–Fialkow criterion in real case, or its hyponormal extension in complex case), which enables explicit extraction of atomic global minimizers (Josz et al., 2017).
• The multi-ordered and symmetry-adapted hierarchies preserve this structure, provided the (blockwise) flatness condition is satisfied on the active moment submatrix (Josz et al., 2017).

7. Open Directions and Future Work

Research frontiers include:

• Robust extraction methods for global minimizers in cases of multiplicity or nonuniqueness.
• Fully adaptive order-selection heuristics with provable performance guarantees.
• Extension to mixed-integer polynomial optimization.
• Further categorization and exploitation of symmetry types beyond torus and sign invariance.
• Applications to more general nonconvex programming domains, including high-dimensional signal inference, algebraic coding, and optimal design for nonlinear systems (Josz et al., 2017).

The Lasserre hierarchy, with its modern extensions and scalable formulations, forms a foundational tool in both theory and practice for global polynomial optimization at scale.