Successive Convex Approximation for Element Placement

• The paper demonstrates that using the AM inequality creates convex upper bounds to relax nonconvex product constraints in element placement problems.
• The SCA framework iteratively updates auxiliary variables to solve a convex surrogate, ensuring monotonic objective decrease and convergence to stationary points.
• The method is validated in quantum source placement tasks, showing convergence in under 20 iterations with improved pairwise entanglement performance.

Successive convex approximation (SCA) for element placement addresses a class of nonconvex optimization problems in which the objective or constraints involve products or ratios of positive functions of the design variables, leading to nonconvex and often NP-hard formulations. By introducing a convex upper-bound via the arithmetic mean (AM) inequality, and iteratively optimizing these surrogates, SCA enables the efficient solution of otherwise intractable element placement problems, such as optimal positioning of quantum network sources under pairwise loss objectives (Qian et al., 2024).

1. Convex Upper Bounds for Multiplicative Terms

Let $f_i(x)>0$ for $i=1,\dots,N$, and define $P(x)=\prod_{i=1}^N f_i(x)$. The arithmetic–geometric mean inequality yields: $$\sqrt[N]{\prod_{i=1}^N f_i(x)} \le \frac{1}{N} \sum_{i=1}^N f_i(x),$$ implying the product admits the convex global upper-bound

$$P(x) \le \left(\frac{1}{N}\sum_{i=1}^N f_i(x)\right)^N.$$

This AM upper-bound is convex with respect to $(f_1(x),\dots,f_N(x))$ when each $f_i$ is convex and positive. Specifically, letting $S(x) = \tfrac{1}{N} \sum_i f_i(x)$ and $h(t) = t^N$ for $t \ge 0$, $h$ is convex and nondecreasing for $N \ge 1$, and thus the composition $x \mapsto h(S(x))$ is convex by the standard composition rule.

The AM upper-bound enables tractable relaxation of nonconvex products arising in element placement and related problems, providing a closed-form convex surrogate amenable to standard optimization tools (Qian et al., 2024).

2. Successive Convex Approximation (SCA) Algorithmic Framework

Consider the general minimization problem: $$\min_{x \in \mathcal{X}} J(x) + \sum_{n=1}^M \prod_{i=1}^{N_n} f_{n,i}(x),$$ where $J(x)$ is convex, each $f_{n,i}(x)$ is positive and sufficiently smooth, and $\mathcal{X}$ is a convex, compact set. Each product is relaxed using its AM upper-bound: $$\prod_{i=1}^{N_n} f_{n,i}(x) \le \left(\frac{1}{N_n} \sum_{i=1}^{N_n} f_{n,i}(x)\right)^{N_n} =: F_n^{\text{AM}}(x).$$ At iteration $k$, the convex surrogate problem is: $$x^{(k+1)} = \arg\min_{x \in \mathcal{X}} J(x) + \sum_{n=1}^M F_n^{\text{AM}}(x).$$ Key conditions ensuring convergence to a stationary point include:

• $\mathcal{X}$ is convex and compact,
• Each $F_n^{\text{AM}}(x)$ is convex in $x$,
• $J(x)$ is convex.

The SCA majorizes the original objective, yielding a non-increasing sequence of objective values and, via standard limit arguments, convergence to stationary points as characterized by KKT conditions (Qian et al., 2024).

3. Application to Quantum Source Placement

The SCA with the AM upper-bound has been specialized to quantum source positioning. The setup features $N$ quantum nodes at fixed positions $\{u_n^{(q)}\}_{n=1}^N$; the design variable is the source location $q \in \mathbb{R}^2$. The objective is to minimize the total pairwise “loss”: $$\min_{q \in \mathbb{R}^2} \sum_{m=1}^M \alpha_m^{-1} 10^{\frac{\eta}{10} (\|q-u_{n_m}\| + \|q-u_{n'_m}\|)} \cdot 10^{\beta |\|q-u_{n_m}\|-\|q-u_{n'_m}\||},$$ where $M$ is the number of node pairs, and $\alpha_m$, $\eta$, $\beta$ are constants.

The problematic term $|\|q-u_n\|-\|q-u_{n'}\||$ leads to a nonconvex product in constraints: $$\|q-u_n\|^2 + \|q-u_{n'}\|^2 - 2\|q-u_n\|\|q-u_{n'}\| \le r_m^2.$$ Applying the AM bound decouples the product: $$\|q-u_n\|\|q-u_{n'}\| \le \frac{1}{2}\left( \|q-u_n\|^2 y_m + \frac{\|q-u_{n'}\|^2}{y_m} \right),$$ with the auxiliary variable $y_m$. The first-order Taylor expansion is used on the right-hand side of the constraint to maintain convexity. At each SCA iteration, $y_m$ is updated to $$y_m^{(k)} = \frac{1}{2} \frac{\|q^{(k)}-u_{n'}\|}{\|q^{(k)}-u_n\|}$$, and $r_m^2$ is linearized around the previous iterate. The resulting problem is a convex program solvable by standard conic or quadratic programming solvers (Qian et al., 2024).

4. Surrogate Problem Construction and Auxiliary Variable Updates

The SCA framework for element placement using the AM upper-bound involves two alternating steps:

1. Tighten surrogate parameters: Update auxiliary variables (such as $y_m$) to locally majorize the nonconvex terms at the current iterate.
2. Solve the convex surrogate: Minimize the majorizing surrogate objective/constraint using a convex solver.

For the quantum source placement example, the constraint at iteration $k$ is: $$\|q-u_n\|^2 + \|q-u_{n'}\|^2 - (\|q-u_n\|^2 y_m^{(k)}+\frac{\|q-u_{n'}\|^2}{y_m^{(k)}}) \le (r_m^{(k)})^2 + 2 r_m^{(k)} (r_m - r_m^{(k)}),$$ which is jointly convex in $(q, r)$. The iterative scheme ensures each surrogate is tangent at the current iterate, preserving tightness.

5. Convergence Properties and Numerical Observations

Empirically, the SCA with AM upper bounds for element placement demonstrates:

• Monotonic decrease of the original (nonconvex) objective value at each iteration.
• Typical convergence in fewer than 20 iterations for quantum source placement instances.
• Superior final positioning, in terms of maximum pairwise entanglement loss, compared with first-order Taylor-based SCA or gradient descent.
• Per-iteration computational costs comparable to other convexified approaches, but with improved objective values and accuracy (Qian et al., 2024).

A plausible implication is that incorporating the AM upper-bound in the SCA framework achieves both theoretical guarantees (stationarity to KKT points) and practical efficiency, outperforming baseline and purely first-order methods in element placement tasks.

6. Summary and Implications for Broader Network Optimization

Successive convex approximation using AM upper bounds provides a principled, efficient technique for decoupling and optimizing nonconvex products and ratios prevalent in network optimization, including various element placement scenarios. The methodology systematically alternates between (i) updating bound-tightening parameters via auxiliary variables and (ii) minimizing convex surrogates in the design variables. This framework not only guarantees convergence to stationary points under standard convexity and compactness assumptions but also enables direct application to practical problems, such as quantum source positioning, with demonstrable numerical advantages over existing methods (Qian et al., 2024).